RBSE Solutions Class 8 Maths Chapter 7 Construction of Quadrilaterals More Ques

I. Objective Type Questions

Question 1. To draw a unique quadrilateral, the total measurements are required
(a) two
(b) three
(c) four
(d) five
Answer: (d) five
In simple words: To draw a unique quadrilateral that looks exactly one way, you need at least five measurements. These measurements help fix the size and shape of the quadrilateral.

🎯 Exam Tip: Remember that knowing only four side lengths is not enough to define a unique quadrilateral, as it can be 'bent' into different shapes; an extra diagonal or angle is needed.

Question 2. A line segment connecting two opposite vertices is called
(a) side
(b) diagonal
(c) hypotenuse
(d) base
Answer: (b) diagonal
In simple words: A diagonal is a straight line that connects two corners of a shape that are not next to each other. It cuts across the middle of the shape.

🎯 Exam Tip: Distinguish carefully between a diagonal (connecting non-adjacent vertices) and a side (connecting adjacent vertices).

Question 3. A parallelogram with sides of equal length is called
(a) triangle
(b) rhombus
(c) rectangle
(d) trapezium
Answer: (b) rhombus
In simple words: A rhombus is a special type of parallelogram where all four sides are the same length. Its opposite angles are also equal.

🎯 Exam Tip: A square is a special type of rhombus that also has all 90-degree angles, so a rhombus is a more general shape.

Question 4. If the lengths of four sides and a diagonal is given then the figure is
(a) triangle
(b) quadrilateral
(c) Processing math: 100%
(d) hexagon
Answer: (b) quadrilateral
In simple words: If you know all four side lengths and one diagonal of a figure, you can draw a unique four-sided shape called a quadrilateral. This information is enough to fix its shape.

🎯 Exam Tip: The diagonal helps to divide the quadrilateral into two triangles, and triangles are uniquely determined by their side lengths.

Question 6. If the lengths of three sides and two diagonals, then the figure is
(a) pentagon
(b) triangle
(c) hexagon
(d) quadrilateral
Answer: (d) quadrilateral
In simple words: When you are given three sides and two diagonals, you have enough information to uniquely build a four-sided shape, which is a quadrilateral. The diagonals help to define the interior structure.

🎯 Exam Tip: Always visualize how the given measurements form triangles, as triangles are the basic building blocks for constructing any polygon.

Question 7. What are the measurements of sides BC and DC in given parallelogram
(a) 9 cm and 2 cm
(b) 3 cm and 6 cm
(c) 4 cm and 5 cm
(d) 7 cm and 7.7 cm
Answer: (c) 4 cm and 5 cm
In simple words: In a parallelogram, opposite sides are always equal in length. So, if AD is 4 cm, then BC will also be 4 cm. If AB is 5 cm, then DC will also be 5 cm. The diagonal length is given, but it is not needed to find the lengths of the opposite sides.

🎯 Exam Tip: Always remember the basic properties of quadrilaterals, especially that opposite sides of a parallelogram are equal and parallel. This property helps solve many problems.

Question 8. Which is not a property of rhombus
(a) all sides equal
(b) diagonals equal
(c) diagonals bisects each other
(d) diagonals are perpendicular to each other
Answer: (b) diagonals equal
In simple words: In a rhombus, all sides are equal, and the diagonals cut each other in half at 90-degree angles. However, the diagonals themselves are not equal in length unless it's also a square. Only rectangles and squares have equal diagonals.

🎯 Exam Tip: Understand the unique properties for each type of quadrilateral (parallelogram, rhombus, rectangle, square, kite, trapezium) to correctly identify them.

II. Fill in the Blanks

Question 1. If the diagonals of a parallelogram are perpendicular to each other than it is a ......
Answer: rhombus
In simple words: If the two lines that cut across a parallelogram meet at a perfect right angle (90 degrees), then that parallelogram must be a rhombus. This is a special characteristic of a rhombus.

🎯 Exam Tip: Remember that a rhombus has perpendicular diagonals, while a rectangle has equal diagonals. A square has both properties.

Question 2. If the diagonals of a quadrilateral bisect each other, then it is a ......
Answer: parallelogram
In simple words: If the two diagonals of a four-sided shape cut each other exactly in half at their meeting point, then that shape is always a parallelogram. This means its opposite sides are parallel.

🎯 Exam Tip: This is a defining property of a parallelogram; if the diagonals bisect each other, it proves the figure is a parallelogram.

Question 3. Each angle of rectangle and square is a ......
Answer: \( 90^\circ \)
In simple words: Every corner (angle) inside a rectangle and a square is a right angle, measuring exactly 90 degrees. This is what makes their corners sharp and straight.

🎯 Exam Tip: Knowing the angle properties helps distinguish rectangles and squares from other quadrilaterals like rhombuses or parallelograms.

Question 4. Opposite sides of a parallelogram are ......
Answer: equal and parallel
In simple words: In a parallelogram, the sides facing each other are always the same length and they run in the same direction without ever meeting (they are parallel). This is a core feature of a parallelogram.

🎯 Exam Tip: Parallel and equal sides are fundamental properties for classifying parallelograms and related figures like rectangles, rhombuses, and squares.

Question 5. A parallelogram with sides of equal length is called ......
Answer: rhombus
In simple words: When all four sides of a parallelogram are of the same length, that special shape is known as a rhombus. It's like a tilted square.

🎯 Exam Tip: Understand that a rhombus is a type of parallelogram, meaning it inherits all parallelogram properties plus its own unique characteristics.

III. Very Short Answer Type Questions

Question 1. Write any two necessary conditions for constructing a unique square.
Answer: A unique square can be constructed if:
• the length of its four sides and a diagonal is given.
• the two diagonals and three sides are known.
In simple words: To draw only one specific square, you need enough information to fix its shape. This can be done if you know all four sides and one diagonal, or if you know both diagonals and three sides. Since a square has equal sides and 90-degree angles, you usually only need one side length or a diagonal to define it fully.

🎯 Exam Tip: For a square, knowing just one side length or one diagonal is enough because all sides are equal and all angles are 90 degrees, simplifying construction conditions.

Question 2. Define square.
Answer: A square is a rectangle with equal sides. It is a special type of quadrilateral where all four sides are equal in length and all four interior angles are right angles (90 degrees).
In simple words: A square is a four-sided shape where all its sides are the same length and all its corners are perfect right angles. It's a special kind of rectangle and rhombus.

🎯 Exam Tip: Remember that a square possesses all the properties of a rectangle, a rhombus, and a parallelogram, making it a highly versatile shape.

Question 3. If two diagonals are equal in a parallelogram, then which type of quadrilateral is this?
Answer: Rectangle.
A rectangle is a parallelogram where the diagonals are equal in length. This property ensures that all interior angles are right angles.
In simple words: If the two diagonals inside a parallelogram are exactly the same length, then that parallelogram is a rectangle. This is how you can tell it has square corners.

🎯 Exam Tip: Keep in mind that for a general parallelogram, the diagonals bisect each other but are not necessarily equal. Equality of diagonals signifies a rectangle.

Question 4. Define rhombus.
Answer: A rhombus is a parallelogram with sides of equal length. Its opposite angles are equal, and its diagonals intersect at right angles.
In simple words: A rhombus is a four-sided shape where all the sides are the same length. It looks like a square that has been pushed over, so its corners are not necessarily 90 degrees.

🎯 Exam Tip: A key property of a rhombus is that its diagonals are perpendicular bisectors of each other, which means they cut each other in half at a 90-degree angle.

Question 5. Define trapezium.
Answer: A trapezium is a quadrilateral with a pair of parallel sides. It is a four-sided polygon that has at least one pair of parallel sides.
In simple words: A trapezium is a four-sided shape that has only one pair of sides that are parallel (meaning they would never meet if extended). The other two sides are not parallel.

🎯 Exam Tip: An isosceles trapezium is a special type where the non-parallel sides are equal, and the base angles are equal.

Question 7. Write any two conditions for constructing a unique quadrilateral.
Answer: A quadrilateral can be constructed uniquely if:
• the length of its four sides and a diagonal is given.
• its two adjacent sides and three angles are known.
In simple words: To draw a quadrilateral that is exactly one specific shape, you need enough information. Two common ways are knowing all four side lengths and one diagonal, or knowing two sides that touch each other and three of its angles.

🎯 Exam Tip: There are five main conditions for constructing a unique quadrilateral: four sides and a diagonal, three sides and two diagonals, two adjacent sides and three angles, three sides and two included angles, or all four sides and one angle. Knowing these helps choose the right method.

IV. Short Answer Type Questions - Think, Discuss And Write

Question 1. Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm, \( \angle A = 50^\circ \), AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique quadrilateral? Give reasons for your answer.
Answer: The quadrilateral ABCD with the given data cannot be constructed because of the following reasons:
• If Arshad takes diagonal BD first, he can construct one triangle, \( \triangle ABD \). But it is not possible to complete the quadrilateral ABCD because only one side, BD, of the second triangle (e.g., \( \triangle BCD \)) is known, and the other necessary measurements for \( \triangle BCD \) (like BC and CD) are missing.
• If Arshad takes the diagonal AC first, then the construction of \( \triangle ACD \) and \( \triangle ABC \) is not possible as the data for these triangles are insufficient. For instance, for \( \triangle ABC \), he has AB = 5 cm, AC = 4 cm, \( \angle A = 50^\circ \). To form a triangle with two sides and one angle, the angle must be included between the two sides, or enough information for another side/angle is needed. Here, \( \angle A \) is not included between AB and AC. A triangle needs at least three specific measurements to be constructed.
In simple words: Arshad cannot draw this quadrilateral uniquely because the measurements he has are not enough to fix the shape. Even if he starts with one diagonal to make a triangle, he doesn't have enough information to make the second triangle needed to complete the quadrilateral. The pieces of information he has don't connect in a way that allows a clear drawing.

🎯 Exam Tip: When constructing quadrilaterals, always check if the given measurements form valid triangles using rules like the triangle inequality (sum of two sides must be greater than the third side) and if angles are included between known sides.

Question 2. We saw that 5 measurements of a quadrilateral can determine a quadrilateral uniquely. Do you think any five measurements of the quadrilateral can do this?
Answer: It is necessary to have at least the knowledge of five parts to be able to construct a quadrilateral uniquely. However, not *any* five parts will work. The specific combination of measurements matters. Here are some combinations that allow for unique construction:
• When four sides and one diagonal are given.
• When two diagonals and three sides are given.
• When two adjacent sides and three angles are given.
In simple words: No, not just any five measurements can make a unique quadrilateral. You need specific types of measurements, like all four sides plus one diagonal, or two diagonals and three sides. The way these five measurements are connected is important to fix the shape so it can only be drawn in one way.

🎯 Exam Tip: Understand the five main conditions for unique quadrilateral construction, as listed in textbooks, to know which combinations of measurements are sufficient.

Question 3. Can you draw a parallelogram BATS where BA = 5 cm, AT = 6 cm and AS = 6.5 cm? Why?
Answer: Yes, the parallelogram BATS with the given data can be drawn. This is because a parallelogram can always be divided into two triangles by a diagonal. Here, the diagonal AS is given, and with sides BA and AT, we can construct \( \triangle BAS \) and \( \triangle ATS \). For \( \triangle BAS \), we have sides BA = 5 cm, AS = 6.5 cm. Since it's a parallelogram, opposite side BT = AS = 6.5 cm and ST = BA = 5 cm. For \( \triangle ATS \), we have AT = 6 cm, TS = 5 cm, and AS = 6.5 cm. Since all three sides of both triangles are known, they can be uniquely constructed.
In simple words: Yes, you can draw this parallelogram. A parallelogram can always be split into two triangles by drawing a line across it (a diagonal). Since you have the lengths of the sides and a diagonal, you have enough information to draw these two triangles, which then form the parallelogram.

🎯 Exam Tip: Any polygon can be broken down into triangles. If you can construct all the component triangles, you can construct the polygon.

Question 4. Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why?
Answer: Yes, the rhombus ZEAL with the given data can be drawn. A rhombus has all four sides equal. So, if ZE = 3.5 cm, then EA = AL = LZ = 3.5 cm. The diagonal EL = 5 cm is also given. This means we have three sides of \( \triangle ZEL \) (ZE=3.5cm, ZL=3.5cm, EL=5cm) and three sides of \( \triangle LEA \) (LE=5cm, EA=3.5cm, AL=3.5cm). Since we have the three sides for each triangle, we can construct both triangles uniquely, which then form the rhombus.
In simple words: Yes, you can draw this rhombus. Because all sides of a rhombus are equal, knowing one side (ZE) means you know all four sides. With one diagonal (EL) also given, you have enough information (three sides for each of the two triangles formed by the diagonal) to draw the whole shape.

🎯 Exam Tip: Remember that a rhombus has four equal sides. This simplifies construction requirements significantly, often needing only one side and a diagonal.

Question 5. A student attempted to draw a quadrilateral PLAY where PL = 3 cm, LA = 4 cm, AY = 4.5 cm, PY = 2 cm and LY = 6 cm, but could not draw it What is the reason? [Hint : Discuss it using a rough sketch].
Answer: A rough sketch is given here according to given data.We observe that for \( \triangle PLY \), the sides are PL = 3 cm, PY = 2 cm, and LY = 6 cm.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In \( \triangle PLY \):
PY + PL \( = 2 + 3 = 5 \) cm.
Since \( 5 \text{ cm} \not> 6 \text{ cm} \) (i.e., PY + PL is not greater than LY), this violates the triangle inequality.
Therefore, the triangle \( \triangle PLY \) cannot be formed with these side lengths. If \( \triangle PLY \) cannot be formed, then the entire quadrilateral PLAY cannot be constructed.
In simple words: The student couldn't draw the shape because one of the triangles inside it can't actually exist. If you try to make a triangle with sides 2 cm, 3 cm, and 6 cm, the two shorter sides (2+3=5 cm) are not long enough to reach across the longest side (6 cm). So, the triangle and thus the whole quadrilateral cannot be made.

🎯 Exam Tip: Always check the triangle inequality theorem for any potential triangles within a polygon. If any triangle cannot be formed, the polygon also cannot be constructed.

Question 6. Can you construct a quadrilateral PQRS with PQ = 3 cm, RS = 3 cm, PS = 7.5 cm, PR = 8 cm and SQ = 4 cm? Justify your answer.
Answer: No, the quadrilateral PQRS cannot be drawn with the given measurements. We can check this by considering the triangles formed by the diagonals.
Let's consider \( \triangle QSP \) with sides PQ = 3 cm, SQ = 4 cm, and PS = 7.5 cm.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In \( \triangle QSP \):
SQ + QP \( = 4 + 3 = 7 \) cm.
We compare this to the third side PS = 7.5 cm.
Since \( 7 \text{ cm} \not> 7.5 \text{ cm} \) (i.e., SQ + QP is not greater than PS), this violates the triangle inequality.
Therefore, the triangle \( \triangle QSP \) cannot be formed with these side lengths, and consequently, the quadrilateral PQRS cannot be constructed.
In simple words: No, you cannot draw this quadrilateral. If you try to make one of the triangles inside it (triangle QSP), two of its sides (3 cm and 4 cm) when added together (7 cm) are not long enough to be greater than the third side (7.5 cm). This means that specific triangle cannot be made, so the whole quadrilateral cannot be built.

🎯 Exam Tip: Always identify the triangles formed by the diagonals within the quadrilateral and apply the triangle inequality theorem to each of them. If any one triangle cannot be formed, the quadrilateral cannot be constructed.

Question 7. Construct a quadrilateral ABCD whose sides are AB = 4 cm, BC = 5 cm, CD = 4.5 cm, \( \angle B = 60^\circ \) and \( \angle C = 90^\circ \).
Answer:(i) First of all we draw a rough sketch of the required quadrilateral and write down the given dimensions.
**Steps of Construction:**
1. Draw a line segment EA = 5 cm. This will be the base for our construction, corresponding to BC in the question but using the solution's labels.
2. At point E, make an angle \( \angle XEA = 60^\circ \) using a protractor and ruler. This angle corresponds to \( \angle B \) from the question.
3. With E as the center and a radius of 4 cm (corresponding to CD in the question), cut off ED = 4 cm along the ray EX.
4. At point A, make an angle \( \angle YAE = 90^\circ \) using a protractor. This angle corresponds to \( \angle C \) from the question.
5. With A as the center and a radius of 4.5 cm (corresponding to AB in the question), draw an arc to cut off AY at R.
6. Finally, join points D and R.
Thus, DEAD is the required quadrilateral.In simple words: To draw this quadrilateral, first draw a base line. Then, from one end of this line, draw another line at the given angle and length. From the other end of the base line, draw a third line at its given angle and length. Finally, connect the last two points to complete the four-sided figure. The specific angles and lengths help make the shape unique.

🎯 Exam Tip: Always start with a rough sketch to visualize the quadrilateral and plan your construction steps. Begin with the side that includes the given angles, if possible, to make it easier.

Question 9. Construct a square ABCD, given that diagonal BD = 5.6 cm.
Answer: First we make a rough diagram for given data.
**Construction Steps:**
1. Draw diagonal AC = 5.6 cm. (Note: The question states BD = 5.6 cm, but the solution proceeds with AC. We follow the solution's steps.)
2. Draw the perpendicular bisector of AC. Let it intersect AC at point O. This bisector ensures that the diagonals of the square will be perpendicular and bisect each other.
3. Since diagonals of a square are equal and bisect each other, OD = OB = \( \frac{1}{2} \) AC = \( \frac{1}{2} \times 5.6 = 2.8 \) cm.
4. With O as the center and a radius of 2.8 cm, draw an arc on both sides of AC along the perpendicular bisector. Mark these points as D and B.
5. Join AD, DC, CB, and BA.
Thus, ABCD is the required square.
In simple words: To make a square when you know one diagonal, first draw that diagonal line. Then, find its exact middle and draw a line perfectly straight up and down through it. On this new line, mark two points that are half the length of the diagonal away from the center. Connect these four points, and you will have your square.

🎯 Exam Tip: Remember that the diagonals of a square are equal, bisect each other, and are perpendicular. This property is key for constructing a square when only a diagonal is given.

Question 10. Construct a rhombus PQRS whose diagonals are PR = 8 cm and QS = 10 cm.
Answer:
**Construction Steps:**
1. First, draw the diagonal PR = 8 cm.
2. Draw the perpendicular bisector of the diagonal PR. This bisector will pass through the midpoint of PR. Let the point where it meets PR be O.
3. Since the diagonals of a rhombus bisect each other perpendicularly, and QS = 10 cm, then OQ = OS = \( \frac{1}{2} \) QS = \( \frac{1}{2} \times 10 = 5 \) cm.
4. With O as the center and a radius of 5 cm, draw arcs on both sides of PR along the perpendicular bisector. Mark the points where these arcs intersect the bisector as Q and S.
5. Join PQ, QR, RS, and SP.
Thus, PQRS is the required rhombus.
In simple words: To draw a rhombus when you know both diagonals, first draw one diagonal. Then, draw a line that cuts this diagonal exactly in half and is perfectly straight up-and-down (perpendicular). From the middle point, measure half of the second diagonal length on both sides along this new line. Connect all four end points, and you will have your rhombus.

🎯 Exam Tip: The critical properties for constructing a rhombus from diagonals are that they bisect each other at right angles. This allows you to set up the framework for the rhombus.

Question 11. Construct a quadrilateral ABCD whose sides are AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and diagonal AC = 7 cm.
Answer: First we draw a rough sketch of quadrilateral ABCD. Draw a diagonal AC and write down its dimension in rough figure.
**Steps of Construction:**
1. First of all, draw a line segment AB = 4.5 cm.
2. With B as the center, draw an arc of 5.5 cm. With A as the center, draw an arc of 7.0 cm. These two arcs will intersect at a point. Label this intersection point C.
3. Join AC and BC. This completes \( \triangle ABC \).
4. With A as the center, draw an arc of 6 cm (for AD).
5. With C as the center, draw an arc of 4 cm (for CD). These two arcs will intersect at a point. Label this intersection point D.
6. Join AD and CD.
Thus, ABCD is the required quadrilateral.
In simple words: To draw this four-sided figure, start by drawing one side. Then, use the given lengths and a diagonal to form the first triangle. After that, use the remaining side lengths and the diagonal again to form the second triangle. Connecting all the points will create the full quadrilateral.

🎯 Exam Tip: This method (four sides and one diagonal) is common. The diagonal divides the quadrilateral into two triangles, which can then be constructed using the SSS (Side-Side-Side) criterion.

Question 12. Construct a rectangle whose adjacent sides are 6 cm and 4 cm.
Answer: First we draw a rough sketch of rectangle ABCD. Write down its dimension in rough figure.
We know that opposite sides of a rectangle are equal and each angle is \( 90^\circ \).
**Steps of Construction:**
1. First, draw a line segment AB = 6 cm.
2. At point A, draw a ray AX such that \( \angle BAX = 90^\circ \). This sets up one of the right angles of the rectangle.
3. With A as the center and a radius of 4 cm (the width), draw an arc that cuts the ray AX at point D.
4. With B as the center and a radius of 4 cm, draw an arc. This will be the side BC.
5. With D as the center and a radius of 6 cm (the length of CD, equal to AB), draw another arc. This arc should intersect the previous arc (from step 4) at point C.
6. Join BC and CD.
Thus, ABCD is the required rectangle.
In simple words: To draw a rectangle with given side lengths, first draw the longest side. At one end of this side, draw a perfectly straight line upwards to make a right angle. Measure the shorter side length along this new line. Do the same for the other end of the first side. Finally, connect the top two points to complete your rectangle.

🎯 Exam Tip: The key properties of a rectangle for construction are that opposite sides are equal and all angles are 90 degrees. Using a protractor for the 90-degree angles is essential.

I. Objective Type Questions

Question 1. To draw a unique quadrilateral, the total measurements are required
(a) two
(b) three
(c) four
(d) five
Answer: (d) five
In simple words: To draw a quadrilateral that is exactly one specific shape and size, you need to know five different measurements, like side lengths or angles. This ensures the quadrilateral cannot be drawn in any other way.

🎯 Exam Tip: Remember that for a unique triangle, you only need three measurements, but for a unique quadrilateral, you need five.

Question 2. A line segment connecting two opposite vertices is called
(a) side
(b) diagonal
(c) hypotenuse
(d) base
Answer: (b) diagonal
In simple words: A diagonal is a straight line that connects two corners of a shape that are not next to each other. It cuts across the inside of the shape.

🎯 Exam Tip: Diagonals are important in quadrilaterals for determining properties like congruence and special types of shapes.

Question 3. A parallelogram with sides of equal length is called
(a) triangle
(b) rhombus
(c) rectangle
(d) trapezium
Answer: (b) rhombus
In simple words: A rhombus is a special type of parallelogram where all four sides are the same length. Think of it like a "tilted square".

🎯 Exam Tip: Remember that a square is a special type of rhombus where all angles are 90 degrees.

Question 4. If the lengths Of four sides and a diagonal is given then the figure is
(a) triangle
(b) quadrilateral
(c) pentagon
(d) hexagon
Answer: (b) quadrilateral
In simple words: If you know all four side lengths and one diagonal of a shape, you can usually draw a quadrilateral. This is enough information to fix its shape.

🎯 Exam Tip: Knowing four sides and a diagonal helps you divide the quadrilateral into two triangles, making it easier to construct.

Question 5.
(b) 360°
(c) 180°
(d) 280°
Answer: (b) 360°
In simple words: The sum of all interior angles in any quadrilateral is always 360 degrees. This is a fundamental property of four-sided shapes.

🎯 Exam Tip: You can quickly check this by dividing any quadrilateral into two triangles; each triangle has 180°, so two triangles make 360°.

Question 6. If the lengths of three sides and two diagonals, then the figure is
(a) pentagon
(b) triangle
(c) hexagon
(d) quadrilateral
Answer: (d) quadrilateral
In simple words: With three sides and two diagonals, you have enough information to uniquely draw a four-sided figure, which is a quadrilateral. The two diagonals help to set the shape of the quadrilateral.

🎯 Exam Tip: Always visualize how the given measurements define the shape; three sides and two diagonals essentially create a network of triangles.

Question 7. What are the measurements of sides BC and DC in given parallelogram
(a) 9 cm and 2 cm
(b) 3 cm and 6 cm
(c) 4 cm and 5 cm
(d) 7 cm and 7.7 cm
Answer: (c) 4 cm and 5 cm
In simple words: In a parallelogram, opposite sides are always equal in length. So, if one side is 4 cm, the opposite side is also 4 cm, and if another side is 5 cm, its opposite side is also 5 cm.

🎯 Exam Tip: Remember the key properties of parallelograms: opposite sides are equal and parallel, and opposite angles are equal.

Question 8. Which is not a property of rhombus
(a) all sides equal
(b) diagonals equal
(c) diagonals bisects each other
(d) diagonals are perpendicular to each other
Answer: (b) diagonals equal
In simple words: In a rhombus, all sides are equal, and the diagonals cut each other in half at a 90-degree angle. However, the diagonals are only equal in length if the rhombus is also a square.

🎯 Exam Tip: A square is a special type of rhombus where the diagonals are also equal; otherwise, in a general rhombus, diagonals are not necessarily equal.

Question 9. Number of diagonals in a quadrilateral are
Answer: (b) 2
In simple words: A quadrilateral is a shape with four sides. You can draw two diagonals inside any quadrilateral by connecting opposite corners.

🎯 Exam Tip: The formula for the number of diagonals in an n-sided polygon is \( \frac{n(n-3)}{2} \). For a quadrilateral, \( n=4 \), so \( \frac{4(4-3)}{2} = \frac{4 \times 1}{2} = 2 \).

II. Fill in the blanks

Question 1. If the diagonals of a parallelogram are perpendicular to each other than it is a .......
Answer: rhombus
In simple words: A parallelogram whose diagonals cross each other at a right angle (90 degrees) is specifically called a rhombus. This makes all its sides equal.

🎯 Exam Tip: This is a key distinguishing property: while all parallelograms have diagonals that bisect each other, only rhombuses have perpendicular diagonals.

Question 2. If the diagonals of a quadrilateral bisect each other, then it is a .......
Answer: parallelogram
In simple words: If the two diagonals of a four-sided shape cut each other exactly in half, then that shape is always a parallelogram. This means its opposite sides are parallel.

🎯 Exam Tip: This is one of the important tests to identify if a given quadrilateral is a parallelogram. If its diagonals bisect each other, it is a parallelogram.

Question 3. Each angle of rectangle and square is a .......
Answer: Answer not provided in source.

🎯 Exam Tip: For fill-in-the-blank questions about properties of shapes, recall all the definitions and common characteristics.

Question 4. Opposite sides of a parallelogram are .......
Answer: Answer not provided in source.

🎯 Exam Tip: Be sure to distinguish between properties of sides (parallel, equal) and properties of angles (equal, supplementary) for different quadrilaterals.

Question 5. A parallelogram with sides of equal length is called .......
Answer: Answer not provided in source.

🎯 Exam Tip: Always pay attention to keywords like "parallelogram with sides of equal length" as they define specific geometric figures.

III. Very Short Answer Type Questions

Question 1. Write any two necessary conditions for constructing a unique square.
Answer: A unique square can be constructed if:
• The length of its four sides and a diagonal is given. Since all sides are equal in a square, one side length is sufficient. Also, diagonals are equal.
• The two diagonals and three sides are known. For a square, knowing just one side or one diagonal is enough, as all sides are equal and diagonals are equal and bisect each other at right angles.
In simple words: To draw only one specific square, you just need to know the length of one side. This is because all sides are equal, and all angles are 90 degrees. You could also just know the length of its diagonal.

🎯 Exam Tip: For constructing special quadrilaterals like a square, fewer measurements are needed because of their inherent properties (equal sides, 90-degree angles, equal and perpendicular diagonals).

Question 2. Define square.
Answer: A square is a rectangle with equal sides. It is also a rhombus with all angles equal to 90 degrees. All four sides of a square are equal in length, and all four internal angles are equal to 90 degrees.
In simple words: A square is a flat shape with four straight sides that are all the same length. All its four corners are perfect right angles (90 degrees).

🎯 Exam Tip: Clearly state both properties: equal sides and right angles, to fully define a square.

Question 3. If two diagonals are equal in a parallelogram, then which type of quadrilateral is this?
Answer: Rectangle.
In simple words: If a parallelogram has diagonals that are the same length, then that parallelogram must be a rectangle. This is a special property that makes all its angles 90 degrees.

🎯 Exam Tip: This is a key property that distinguishes a rectangle from other parallelograms; its diagonals are equal in length.

Question 4. Define rhombus.
Answer: A rhombus is a parallelogram with sides of equal length. All four sides of a rhombus are equal, but its angles are not necessarily 90 degrees.
In simple words: A rhombus is a four-sided shape where all four sides are exactly the same length. It looks like a diamond or a square that has been pushed over.

🎯 Exam Tip: Emphasize "all sides equal" and "parallelogram" in your definition to be complete.

Question 5. Define trapezium.
Answer: A trapezium is a quadrilateral with a pair of parallel sides. Only one pair of opposite sides in a trapezium is parallel.
In simple words: A trapezium is a four-sided shape where only two of its sides are parallel to each other. The other two sides are not parallel.

🎯 Exam Tip: The critical feature of a trapezium is that it has *exactly one* pair of parallel sides. If it had two, it would be a parallelogram.

Question 6. W_Processing math: 100% parallelogram.
Answer: Answer not provided in source.

🎯 Exam Tip: When a question is incomplete, focus on understanding the likely topic and recalling related definitions.

Question 7. Write any two conditions for constructing a unique quadrilateral.
Answer: A quadrilateral can be constructed uniquely under the following conditions:
• If the length of its four sides and a diagonal is given. This allows the quadrilateral to be split into two triangles, each of which can be uniquely constructed.
• If its two adjacent sides and three angles are known. The angles help to define the shape and orientation of the sides.
In simple words: To draw a quadrilateral that is exactly one specific shape, you either need to know all four side lengths and one diagonal. Or, you need to know two sides next to each other and three of its angles.

🎯 Exam Tip: Always remember the minimum information needed for unique construction: 5 parts for a general quadrilateral, fewer for special ones like squares or rectangles.

IV. Short Answer Type Questions- Think, Discuss and Write

Question 1. Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm, \( \angle A = 50^\circ \), AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique quadrilateral? Give reasons for your answer.
Answer: The quadrilateral ABCD with the given data cannot be constructed because of the following reasons:
• If Arshad takes diagonal BD first, he can construct one triangle, \( \triangle ABD \). However, the second part to complete the quadrilateral (ABCD) cannot be drawn because only one side (BD) of the second triangle is known. The other needed values are missing.
• If he takes the diagonal AC first, then the construction of triangles \( \triangle ACP \) and \( \triangle ABC \) (likely meant \( \triangle ADC \) and \( \triangle ABC \)) is not possible as the data provided is insufficient to form these triangles uniquely.
In simple words: Arshad cannot draw a unique quadrilateral with these measurements. Even though he has five measurements, they don't give enough information to build clear triangles. For example, if he draws one diagonal, he cannot complete the other part of the quadrilateral.

🎯 Exam Tip: For quadrilateral construction, the five measurements must be *appropriately chosen* to form unique triangles. Not any five measurements will work.

Question 2. We saw that 5 measurements of a quadrilateral can determine a quadrilateral uniquely. Do you think any five measurements of the quadrilateral can do this?
Answer: No, not any five measurements of a quadrilateral can determine it uniquely. It is necessary to have at least the knowledge of five specific parts in a particular combination to be able to construct a quadrilateral uniquely.
• For instance, when four sides and one diagonal are given, a unique quadrilateral can be constructed. This divides the quadrilateral into two triangles that can be constructed.
• Similarly, when two diagonals and three sides are given, a unique quadrilateral can often be constructed. The way the five measurements are chosen is very important.
In simple words: No, just any five measurements are not enough to draw a specific quadrilateral. The measurements must be chosen in a special way, like all four sides and one diagonal, or two diagonals and three sides. The right combination of information is key.

🎯 Exam Tip: Emphasize that "any five measurements" is not enough. The measurements must form valid triangles and define the overall shape without ambiguity.

Question 3. Can you draw a parallelogram BATS where BA = 5 cm, AT = 6 cm and AS = 6.5 cm? Why?
Answer: Yes, the parallelogram BATS with the given data can be drawn. This is because a parallelogram can be divided into two triangles, for example, \( \triangle BAS \) and \( \triangle SAT \). We know three sides of \( \triangle BAS \) (BA, AS, and BS, where BS = AT = 6 cm due to parallelogram properties). Once these are constructed, the parallelogram can be completed.
In simple words: Yes, you can draw this parallelogram. Even though you only have three side lengths, because it's a parallelogram, we know opposite sides are equal. This means you actually have enough information to create two triangles that form the parallelogram.

🎯 Exam Tip: Always use the specific properties of the given quadrilateral (e.g., opposite sides equal in a parallelogram) to find missing measurements needed for construction.

Question 4. Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why?
Answer: Yes, the rhombus ZEAL with the given data can be drawn. A rhombus can be divided into two triangles, \( \triangle ZEL \) and \( \triangle LEA \). Since it is a rhombus, all sides are equal, so \( ZE = EA = AL = LZ = 3.5 \) cm. With \( ZE = 3.5 \) cm, \( ZL = 3.5 \) cm, and \( EL = 5 \) cm, \( \triangle ZEL \) can be constructed. Similarly, \( \triangle LEA \) can be constructed using \( LE = 5 \) cm, \( EA = 3.5 \) cm, and \( AL = 3.5 \) cm.
In simple words: Yes, you can draw this rhombus. Because all sides of a rhombus are equal, knowing one side (ZE) means you know all four sides. Then, with one diagonal (EL), you have enough information to build two triangles that make up the rhombus.

🎯 Exam Tip: For rhombus construction, remember that all four sides are equal, and the diagonals bisect each other at 90 degrees. One side and one diagonal provide enough information.

Question 5. A student attempted to draw a quadrilateral PLAY where PL = 3 cm, LA = 4 cm, AY = 4.5 cm, PY = 2 cm and LY = 6 cm, but could not draw it What is the reason? [Hint : Discuss it using a rough sketch].
Answer: A rough sketch is given here according to given data. We observe that the triangle inequality rule is violated. Specifically, for triangle PLY, the sum of two sides (PY + PL) should be greater than the third side (LY). However, in this case, \( PY + PL = 2 + 3 = 5 \) cm, which is not greater than \( LY = 6 \) cm. Since \( 5 \not> 6 \), the triangle PLY cannot be formed. Thus, the quadrilateral PLAY cannot be drawn.

In simple words: The student could not draw the quadrilateral because one of the triangles (PLY) that would make up the quadrilateral cannot actually exist. This is because two of its sides (2 cm and 3 cm) add up to only 5 cm, which is shorter than the third side (6 cm). A triangle can only be formed if the sum of any two sides is greater than the third side.

🎯 Exam Tip: Always check the triangle inequality theorem (sum of any two sides of a triangle must be greater than the third side) when given side lengths, as a quadrilateral is fundamentally composed of triangles.

Question 6. Can you construct a quadrilateral PQRS with PQ = 3 cm, RS = 3 cm, PS = 7.5 cm, PR = 8 cm and SQ = 4 cm? Justify your answer.
Answer: No, a quadrilateral PQRS with the given measurements cannot be drawn. This is because if we draw a rough sketch and consider triangle QSP, the triangle inequality rule is violated. Specifically, \( SQ + QP \not> SP \). Here, \( SQ + QP = 4 + 3 = 7 \) cm. However, \( SP = 7.5 \) cm. Since \( 7 \not> 7.5 \), the triangle QSP cannot be formed, making the construction of the quadrilateral impossible.
In simple words: No, you cannot draw this quadrilateral. If you try to make a triangle with sides SQ (4 cm), QP (3 cm), and SP (7.5 cm), it won't work. The two shorter sides (4+3=7) are not longer than the longest side (7.5), which is a rule for making any triangle.

🎯 Exam Tip: When constructing any polygon, always check that all internal triangles satisfy the triangle inequality theorem.

Question 7. Construct a quadrilateral ABCD whose sides are AB = 4 cm, BC = 5 cm, CD = 4.5 cm, \( \angle B = 60^\circ \) and \( \angle C = 90^\circ \).
Answer: (Note: The solution steps use labels 'E', 'A', 'D', 'R', and values that don't match the question 'ABCD'. The construction will follow the provided solution's labels and steps for a quadrilateral 'DEAD'.)
(i) First of all, we draw a rough sketch of the required quadrilateral DEAD and write down the given dimensions based on the solution steps.

Steps of Construction:
1. First, draw a line segment \( EA = 5 \) cm.
2. Next, construct \( \angle XEA = 60^\circ \) at point E.
3. With E as the center and a radius of 4 cm, draw an arc that cuts the line \( EX \) at point D. So, \( ED = 4 \) cm.
4. Now, construct \( \angle EAY = 90^\circ \) at point A.
5. With A as the center and a radius of 4.5 cm, draw an arc that cuts the line \( AY \) at point R.
6. Finally, join points D and R. Thus, DEAD is the required quadrilateral.
In simple words: To build this shape, first draw a line segment 5 cm long and label its ends E and A. At E, make a 60-degree angle. Along this angle line, measure 4 cm and mark point D. At A, make a 90-degree angle. Along this new angle line, measure 4.5 cm and mark point R. Then, connect D and R. This forms the quadrilateral DEAD.

🎯 Exam Tip: When constructing quadrilaterals with angles, always draw the base and then the angles first. The sides are then measured along the angle rays. Carefully label points to avoid confusion.

Question 9. Construct a square ABCD, given that diagonal BD = 5.6 cm.
Answer: (Note: The question states diagonal BD = 5.6 cm, but the solution provided constructs with AC = 5.6 cm. Following the solution's construction, which implies diagonal AC = 5.6 cm.)
First, we make a rough diagram for the given data.

Construction Steps:
(i) Draw the diagonal \( AC = 5.6 \) cm.
(ii) Draw the perpendicular bisector of \( AC \), and let it intersect \( AC \) at point O. This bisector is the other diagonal.

(iii) Since diagonals of a square bisect each other perpendicularly and are equal, set your compass to \( 2.8 \) cm (half of 5.6 cm). With O as the center, draw arcs of \( 2.8 \) cm on the perpendicular bisector above and below O, to get points B and D.
(iv) Join AD, DC, CB, and BA. This completes the required square ABCD.
In simple words: To make the square, first draw a line 5.6 cm long and call it AC. Find its exact middle point, O, and draw a straight line through O that is perfectly perpendicular to AC. Now, measure 2.8 cm (half of 5.6) from O along this new line on both sides, and mark points B and D. Connect A, B, C, and D with straight lines. This will form your square.

🎯 Exam Tip: Remember that in a square, diagonals are equal, bisect each other, and are perpendicular. These properties are crucial for its construction using only one diagonal.

Question 10. Construct a rhombus PQRS whose diagonals are PR = 8 cm and QS = 10 cm.
Answer: First, we make a rough diagram for the given data.

Construction Steps:
Step 1. First, draw the diagonal \( PR = 8 \) cm.
Step 2. Draw the perpendicular bisector of the diagonal \( PR \). Let this bisector meet \( PR \) at point O. This line will contain the other diagonal.

Step 3. Since the diagonals of a rhombus bisect each other, half of \( QS \) is \( \frac{10}{2} = 5 \) cm. Taking O as the center, draw arcs of 5 cm length on both sides of O along the perpendicular bisector. Mark these points as Q and S.
Step 4. Join PQ, QR, RS, and SP. This forms the required rhombus PQRS.
In simple words: First, draw a line PR which is 8 cm long. Find its middle point, O, and draw a straight line through O that is perfectly perpendicular to PR. This perpendicular line will hold the other diagonal. Since the other diagonal QS is 10 cm, measure 5 cm (half of 10 cm) from O along this perpendicular line, both up and down, to mark points Q and S. Finally, connect points P, Q, R, S to form the rhombus.

🎯 Exam Tip: When constructing a rhombus using diagonals, remember that diagonals bisect each other at right angles. This allows you to construct the shape by finding the midpoint and using half-lengths of the diagonals.

Question 11. Construct a quadrilateral ABCD whose sides are AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and diagonal AC = 7 cm.
Answer: First, we draw a rough sketch of quadrilateral ABCD and write down its dimensions in the figure.

Steps of Construction:
(i) First of all, draw a line segment \( AB = 4.5 \) cm.
(ii) With B as the center, draw an arc of \( 5.5 \) cm. With A as the center, draw an arc of \( 7.0 \) cm. These two arcs will intersect. Mark the intersection point as C. Join AC and BC.
(iii) With A as the center, draw an arc of \( 6 \) cm. With C as the center, draw an arc of \( 4 \) cm. These two arcs will intersect. Mark this intersection point as D. Join AD and CD. Thus, ABCD is the required quadrilateral.
In simple words: Start by drawing a line AB, 4.5 cm long. From point B, draw an arc with a compass set to 5.5 cm. From point A, draw another arc with a compass set to 7 cm. The point where these two arcs meet is C. Connect A to C and B to C. Now, from point A, draw an arc with radius 6 cm. From point C, draw another arc with radius 4 cm. The point where these arcs cross is D. Connect A to D and C to D to complete the quadrilateral.

🎯 Exam Tip: When constructing a quadrilateral with sides and a diagonal, always construct the two triangles that make up the quadrilateral separately. The diagonal forms a common side for both triangles.

Question 12. Construct a rectangle whose adjacent sides are 6 cm and 4 cm.
Answer: First, we draw a rough sketch of rectangle ABCD and write down its dimensions in the figure.

We know that opposite sides of a rectangle are equal and each angle is \( 90^\circ \).
Steps of Construction:
(i) First, draw a line segment \( AB = 6 \) cm.

(ii) At point A, construct \( \angle BAX = 90^\circ \). Draw an arc of 4 cm with point A as the center, cutting AX at D. So, \( AD = 4 \) cm.
(iii) At point B, construct \( \angle ABY = 90^\circ \). Draw an arc of 4 cm with point B as the center, cutting BY at C. So, \( BC = 4 \) cm.
(iv) Join CD. This forms the required rectangle ABCD.
In simple words: First, draw a 6 cm line and label it AB. At point A, draw a line straight up (at a 90-degree angle) and measure 4 cm along it, marking point D. Do the same at point B: draw a line straight up and measure 4 cm, marking point C. Finally, connect points D and C to complete the rectangle.

🎯 Exam Tip: For constructing a rectangle, remember that all angles are 90 degrees and opposite sides are equal. Constructing two adjacent sides and their right angles is key.