Samacheer Kalvi Class 8 Maths Solutions Chapter 5 Geometry InText Questions

Tamilnadu Samacheer Kalvi 8th Maths Solutions Chapter 5 Geometry InText Questions

Answer the following questions by recalling the properties of triangles:

Question 1. The sum of the three angles of a triangle is 180°.
Answer: The sum of the three angles inside any triangle is always \( 180^\circ \). This is a fundamental property of all triangles.
In simple words: If you add up all the corners of a triangle, you will always get 180 degrees.

🎯 Exam Tip: Remember this property as it's key for solving many geometry problems involving triangles.

Question 2. The exterior angle of a triangle is equal to the sum of the interior opposite angles to it.
Answer: An exterior angle of a triangle is formed when one side is extended. This angle is always equal to the total of the two angles inside the triangle that are not next to it. This property is very useful for finding unknown angles.
In simple words: The angle outside a triangle is the same as adding the two inside angles that are not its neighbours.

🎯 Exam Tip: Clearly identify the exterior angle and its two non-adjacent interior angles before applying this rule.

Question 3. In a triangle, the sum of any two sides is greater than the third side.
Answer: In any triangle, if you add the lengths of any two sides, their sum will always be more than the length of the third side. This is called the Triangle Inequality Theorem and determines if a triangle can actually be formed.
In simple words: If you pick any two sides of a triangle and add their lengths, they will always be longer than the third side.

🎯 Exam Tip: This rule helps check if a given set of three lengths can actually form a triangle.

Question 4. Angles opposite to equal sides are Equal and vice - versa.
Answer: In a triangle, if two sides have the same length, then the angles across from those sides will also be equal. The opposite is also true: if two angles in a triangle are equal, then the sides opposite those angles must also be equal in length. This is a property of isosceles triangles.
In simple words: If two sides of a triangle are the same length, the angles across from them are also the same. And if two angles are the same, the sides across from them are also the same.

🎯 Exam Tip: This is a defining characteristic of an isosceles triangle, so look for equal sides to identify equal angles, and vice-versa.

Question 5. What is \( \angle A \) in the triangle ABC?
Answer: We use the property that the exterior angle of a triangle is equal to the sum of its interior opposite angles.
From the figure, the exterior angle at B is \( 150^\circ \).
We are given \( \angle C = 40^\circ \).
The exterior angle at B \( = \angle A + \angle C \)
\( 150^\circ = \angle A + 40^\circ \)
To find \( \angle A \), we subtract \( 40^\circ \) from \( 150^\circ \).
\( \angle A = 150^\circ - 40^\circ \)
\( \angle A = 110^\circ \). An exterior angle is formed by extending one side and is useful in calculating unknown angles.
In simple words: We know the outside angle is \( 150^\circ \) and one inside angle is \( 40^\circ \). So, we just subtract \( 40^\circ \) from \( 150^\circ \) to find the missing angle \( \angle A \), which is \( 110^\circ \).

🎯 Exam Tip: Clearly label the exterior and interior opposite angles on your diagram to avoid confusion during calculation.

Try These (Text Book page No. 157)

Identify the pairs of figures which are similar and congruent and write the letter pairs.

Answer:
Similar shapes:
(i) W and L
(ii) B and J
(iii) A and G
(iv) B and J
(v) B and Y
(vi) E and N
(vii) H and Q
(viii) R and T
(ix) S and T
Congruent shapes:
(i) Z and I
(ii) J and Y
(iii) C and P
(iv) B and K
(v) R and S
(vi) I and Z
In simple words: Similar shapes have the same form but can be different sizes, like a small photo and a large photo of the same thing. Congruent shapes are exactly the same size and the same form, like two identical copies.

🎯 Exam Tip: Remember that all congruent figures are similar, but not all similar figures are congruent. Congruence means an exact match in both shape and size.

Try These (Text Book page No. 158)

Question 1. Match the following by their congruence property

🎯 Exam Tip: Familiarize yourself with all congruence rules (SSS, SAS, ASA, RHS) and the specific markings that indicate each one on a diagram.

Think (Text Book page No. 160)

Question 1. In the figure, DA = DC and BA = BC. Are the triangles DBA and DBC congruent? Why?
Answer: Yes, triangles DBA and DBC are congruent.
Here's why:
1. We are given that \( AD = CD \) (Side).
2. We are given that \( AB = CB \) (Side).
3. The side \( DB = DB \) is common to both triangles (Side).
Since all three corresponding sides of the two triangles are equal, they are congruent by the Side-Side-Side (SSS) congruence rule. If they are right-angled triangles, they could also be congruent by the Right-angle-Hypotenuse-Side (RHS) rule if the right angle and hypotenuse were specifically marked.
In simple words: The two triangles are exactly the same because all three of their sides match up perfectly with each other. This is called the SSS rule.

🎯 Exam Tip: Always list out the corresponding equal parts (sides or angles) clearly to justify congruence using a specific rule like SSS, SAS, ASA, or RHS.

Activity (Text Book page No. 169)

Question 1. We can construct sets of Pythagorean triplets as follows. Let m and n be any two positive integers (m > n): (a, b, c) is a Pythagorean triplet if \( a = m^2 – n^2 \), \( b = 2mn \) and \( c = m^2 + n^2 \) (Think, why?) Complete the table.

🎯 Exam Tip: This formula \( a = m^2 - n^2 \), \( b = 2mn \), \( c = m^2 + n^2 \) is a general way to generate Pythagorean triplets, ensuring \( m > n \) helps avoid zero or negative side lengths.

Question 2. Find all integer-sided right angled triangles with hypotenuse 85.
Answer: The Pythagorean triplets with a hypotenuse of 85 are:
(13, 84, 85)
(36, 77, 85)
(40, 75, 85)
(51, 68, 85)
These sets of three whole numbers satisfy the Pythagorean theorem \( a^2 + b^2 = c^2 \), where c is 85. Finding these involves factoring 85 and using properties of Pythagorean triplets.
In simple words: We are looking for groups of three whole numbers that make a right-angled triangle, where the longest side (hypotenuse) is 85. We list all such combinations.

🎯 Exam Tip: To find all triplets for a given hypotenuse, you can use the formulas \( m^2 - n^2 \), \( 2mn \), \( m^2 + n^2 \) and look for pairs of m and n such that \( m^2 + n^2 \) equals the given hypotenuse.

Think (Text Book page No. 173)

Question 1. In any acute angled triangle, all three altitudes are inside the triangle. Where will be the orthocentre? In the interior of the triangle or in its exterior?
Answer: In an acute-angled triangle, all three altitudes (the lines drawn from a vertex perpendicular to the opposite side) fall inside the triangle. Therefore, the orthocentre, which is the point where these altitudes meet, will be in the interior of the triangle. The orthocentre's position helps classify triangles by angle.
In simple words: For a triangle with all sharp angles (acute), the lines that show its height will all meet inside the triangle. So, the orthocentre (where they meet) is also inside.

🎯 Exam Tip: The location of the orthocentre (inside, on, or outside the triangle) is directly related to the type of triangle (acute, right, or obtuse, respectively).

Question 2. In any right angled triangle, the altitude perpendicular to the hypotenuse is inside the triangle; the other two altitudes are the legs of the triangle. Can you identify the orthocentre in this case?
Answer: In a right-angled triangle, two of the altitudes are actually the legs (the sides that form the right angle) of the triangle. These two altitudes meet at the vertex where the right angle is located. The third altitude, from the right angle vertex to the hypotenuse, also passes through this vertex. Therefore, the orthocentre is at the vertex containing the \( 90^\circ \) angle. This is a special property of right triangles.
In simple words: For a right-angled triangle, the point where all the height lines meet (the orthocentre) is exactly at the corner where the right angle (90 degrees) is.

🎯 Exam Tip: For right-angled triangles, the orthocentre is always one of the vertices – specifically, the vertex with the right angle.

Question 3. In any obtuse angled triangle, the altitude connected to the obtuse vertex is inside the triangle, and the two altitudes connected to the acute vertices are outside the triangle. Can you identify the orthocentre in this case?
Answer: In an obtuse-angled triangle (a triangle with one angle greater than \( 90^\circ \)), the orthocentre is located outside the triangle. This happens because two of the altitudes fall outside the triangle, requiring the triangle's sides to be extended to meet them. When the three altitude lines (or their extensions) intersect, that point is the orthocentre.
In simple words: For a triangle with one wide (obtuse) angle, the lines for height will cross each other outside the triangle. So, the orthocentre will be outside the triangle too.

🎯 Exam Tip: Visually extend the sides of an obtuse triangle to find where the altitudes from the acute angles would meet, thus locating the orthocentre outside the triangle.

Try These (Text Book page No. 177)

Identify the type of segment required in each triangle: (median, altitude, perpendicular bisector, angle bisector)

Question. (i)
Answer: \( AD = \) Altitude
In simple words: The line AD is drawn from vertex A to the opposite side BC, meeting it at a 90-degree angle, which means it is an altitude.

🎯 Exam Tip: An altitude is always a perpendicular line from a vertex to the opposite side, forming a right angle.

Question. (ii)
Answer: \( l_1 = \) perpendicular bisector
In simple words: The line \( l_1 \) cuts the side BC exactly in half and also forms a 90-degree angle with it, which means it is a perpendicular bisector.

🎯 Exam Tip: A perpendicular bisector both halves a side and meets it at a right angle. It does not necessarily pass through the opposite vertex.

Question. (iii)
Answer: \( BD = \) Median
In simple words: The line BD connects vertex B to the middle point D of the opposite side AC, dividing it into two equal parts (3.5 cm each). This type of line is called a median.

🎯 Exam Tip: A median always connects a vertex to the midpoint of the opposite side, effectively bisecting that side.

Question. (iv)
Answer: \( CD = \) Angular bisector
In simple words: The line CD divides the angle C into two equal angles of \( 25^\circ \) each. This type of line, which splits an angle into two equal parts, is known as an angular bisector.

🎯 Exam Tip: An angle bisector always divides an angle into two equal halves. Look for equal angle markings to identify it.

Think (Text Book page No. 187)

Question 1. Is it possible to construct a quadrilateral PQRS with PQ = 5 cm, QR = 3cm, RS = 6cm, PS = 7cm and PR = 10cm. If not,why?
Answer: No, it is not possible to construct such a quadrilateral.
Let's consider the triangle PQR within the quadrilateral. We have sides:
\( PQ = 5 \text{ cm} \)
\( QR = 3 \text{ cm} \)
\( PR = 10 \text{ cm} \)
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.
For triangle PQR:
\( PQ + QR = 5 + 3 = 8 \text{ cm} \)
However, \( PR = 10 \text{ cm} \).
Since \( PQ + QR = 8 \text{ cm} < 10 \text{ cm} \), the condition for forming a triangle is not met. Thus, the triangle PQR cannot be constructed. Since the triangle PQR cannot be formed, the quadrilateral PQRS also cannot be constructed. This theorem is crucial for geometric constructions.
In simple words: You cannot draw a triangle if two of its sides, when added together, are shorter than the third side. Here, 5 cm plus 3 cm is 8 cm, which is less than the third side of 10 cm. So, the bottom part of the shape cannot be made, meaning the whole shape cannot be made.

🎯 Exam Tip: Always check the triangle inequality theorem for any triangle formed within a polygon before attempting to construct it. The sum of any two sides must be strictly greater than the third side.

Try These (Text Book page No. 187)

Question 1. The area of the trapezium is \( \frac { 1 }{ 2 } \times h \times (a + b) \) sq. units.
Answer: The formula for the area of a trapezium is \( \frac { 1 }{ 2 } \times h \times (a + b) \) square units. Here, 'h' is the height, and 'a' and 'b' are the lengths of the two parallel sides. This formula is derived by dividing the trapezium into simpler shapes like triangles and rectangles.
In simple words: To find the space inside a trapezium, you take half of its height and multiply it by the sum of its two parallel sides.

🎯 Exam Tip: Remember that 'a' and 'b' in the formula refer specifically to the lengths of the parallel sides, and 'h' is the perpendicular distance between them.

Question 2. The distance between the parallel sides of a trapezium is called as its height.
Answer: The distance between the parallel sides of a trapezium is correctly identified as its height. The height is always measured as the perpendicular distance between these two parallel bases. It is crucial to use the perpendicular distance for accurate area calculations.
In simple words: The space from one parallel side to the other, measured straight up and down, is called the height of the trapezium.

🎯 Exam Tip: Always ensure you use the perpendicular distance between the parallel sides when identifying or calculating the height of a trapezium.

Question 3. If the height and parallel sides of a trapezium are 5cm, 7cm and 5cm respectively, then its area is 30.
Answer: Given:
Height \( h = 5 \text{ cm} \)
Parallel side \( a = 7 \text{ cm} \)
Parallel side \( b = 5 \text{ cm} \)
The formula for the area of a trapezium is \( \frac { 1 }{ 2 } \times h \times (a + b) \).
Substitute the values:
Area \( = \frac { 1 }{ 2 } \times 5 \times (7 + 5) \)
Area \( = \frac { 1 }{ 2 } \times 5 \times 12 \)
Area \( = \frac { 1 }{ 2 } \times 60 \)
Area \( = 30 \text{ sq. cm} \). This calculation demonstrates the practical application of the trapezium area formula.
In simple words: We use the formula for a trapezium's area: half times the height times the sum of the two parallel sides. So, it's \( \frac{1}{2} \times 5 \times (7 + 5) \), which gives us \( 30 \text{ sq. cm} \).

🎯 Exam Tip: Clearly write down the given values and the formula before substituting to minimize calculation errors.

Question 4. In an isosceles trapezium, the non-parallel sides are equal in length.
Answer: In an isosceles trapezium, the non-parallel sides are indeed equal in length. This special property distinguishes it from other trapeziums and gives it certain symmetrical characteristics. This is similar to how an isosceles triangle has two equal sides.
In simple words: For a special type of trapezium called an isosceles trapezium, the two sides that are not parallel are the same length.

🎯 Exam Tip: Remember that in an isosceles trapezium, not only are the non-parallel sides equal, but the base angles (angles on the same parallel base) are also equal.

Question 5. To construct a trapezium, Four measurements are enough.
Answer: To successfully construct a trapezium, typically four independent measurements are enough. These could be the lengths of its two parallel sides, its height, and one non-parallel side, or other combinations. Knowing at least four distinct parameters helps define its unique shape.
In simple words: You usually need four pieces of information, like side lengths or angles, to draw a trapezium correctly.

🎯 Exam Tip: Understand which specific four measurements are needed for different construction methods of a trapezium (e.g., two parallel sides, height, and one diagonal).

Question 6. If the area and sum of the parallel sides are 60 cm² and 12 cm, its height is 10 cm.
Answer: Given:
Area of trapezium \( = 60 \text{ cm}^2 \)
Sum of parallel sides \( (a + b) = 12 \text{ cm} \)
The formula for the area of a trapezium is \( \text{Area} = \frac { 1 }{ 2 } \times h \times (a + b) \).
Substitute the given values into the formula:
\( 60 = \frac { 1 }{ 2 } \times h \times 12 \)
Now, simplify the equation to find 'h':
\( 60 = h \times 6 \)
To find h, divide 60 by 6:
\( h = \frac { 60 }{ 6 } \)
\( h = 10 \text{ cm} \). This calculation shows how to work backwards from the area to find the height.
In simple words: We know the area and the total length of the two parallel sides. We can use the area formula to find the height by rearranging it, which gives us \( 10 \text{ cm} \).

🎯 Exam Tip: When solving for an unknown variable in a formula, write down the formula first, substitute known values, and then perform algebraic manipulations carefully.

Activity (Text Book page No. 193 & 194)

Question 1. A pair of identical 30° - 60° - 90° set-squares are needed for this activity. Place them as shown in the figure.
[Image of set-squares forming a parallelogram]
(i) What is the shape we get? It is a parallelogram.
Answer: The shape formed is a parallelogram. When two identical 30-60-90 set-squares are placed as shown, their hypotenuses align and the shorter legs form parallel sides, resulting in a four-sided figure with opposite sides parallel.
In simple words: When you put two of these special triangle rulers together like the picture, you get a four-sided shape where opposite sides run in the same direction, which is called a parallelogram.

🎯 Exam Tip: A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. Recognizing this is key.

Question 1. (ii) Are the opposite sides parallel?
Answer: Yes, the opposite sides are parallel. This is a defining characteristic of a parallelogram, which is formed by the arrangement of the set-squares. The construction naturally leads to this property.
In simple words: Yes, the sides facing each other are parallel, meaning they never meet.

🎯 Exam Tip: Parallel sides are indicated by arrows on diagrams and are a fundamental property of parallelograms and related quadrilaterals.

Question 1. (iii) Are the opposite sides equal?
Answer: Yes, the opposite sides are equal. Since identical set-squares are used, the corresponding sides that form the parallelogram will be equal in length. This is another key property of parallelograms.
In simple words: Yes, the sides that are opposite to each other are the same length.

🎯 Exam Tip: In any parallelogram, not only are opposite sides parallel, but they are also equal in length.

Question 1. (iv) Are the diagonals equal?
Answer: No, the diagonals are not equal. In a general parallelogram, the diagonals are typically not of the same length, unless it's a special type like a rectangle or square. The set-squares form a parallelogram that is not a rectangle, so its diagonals will differ.
In simple words: No, the lines going from one corner to the opposite corner (diagonals) are not the same length in this shape.

🎯 Exam Tip: Diagonals are equal only in specific parallelograms like rectangles and squares; for a general parallelogram, they are unequal but bisect each other.

Question 1. (v) Can you get this shape by using any other pair of identical set-squares?
Answer: Yes, you can get a parallelogram using any other pair of identical set-squares. The fundamental principle is that using two identical triangles to form a quadrilateral will always result in a parallelogram, as long as corresponding sides are joined appropriately. For instance, two identical right-angled isosceles triangles can also form a parallelogram or even a square.
In simple words: Yes, you can make a parallelogram with any two identical set-squares, not just the 30-60-90 ones.

🎯 Exam Tip: The ability to form a parallelogram from two identical triangles highlights the properties of congruence and symmetry in geometric constructions.

Question 2. We need a pair of 30° - 60° - 90° set- squares for this activity. Place them as shown in the figure.
[Image of set-squares forming a rectangle]
(i) What is the shape we get'?
Answer: Rectangle. When the two identical 30-60-90 set-squares are placed in this specific way, they form a four-sided figure with four right angles, which is a rectangle. This arrangement uses the longest side of one set-square aligned with the shortest side of the other.
In simple words: When these specific triangle rulers are put together in this way, you get a rectangle, which is a shape with four straight sides and four square corners.

🎯 Exam Tip: A rectangle is a special type of parallelogram where all angles are \( 90^\circ \). Its opposite sides are parallel and equal.

Question 2. (ii) Is it a parallelogram?
Answer: Yes, it is a parallelogram. A rectangle is a special type of parallelogram because it fulfills all the conditions of a parallelogram: both pairs of opposite sides are parallel and equal. Rectangles simply add the condition of having four right angles.
In simple words: Yes, a rectangle is a type of parallelogram because its opposite sides are parallel.

🎯 Exam Tip: All rectangles are parallelograms, but not all parallelograms are rectangles. Understand the hierarchy of quadrilaterals.

Question 2. (iii) It is a quadrilateral; infact it is a rectangle. (How?)
Opposite sides are equal. All angles = 90°
Answer: The shape is a quadrilateral, and it is a rectangle because its opposite sides are equal in length, and all its interior angles are \( 90^\circ \). The construction with the set-squares creates these specific properties naturally. The 30-60-90 triangles help form the right angles easily.
In simple words: It is a rectangle because its opposite sides are the same length, and every corner is a perfect 90-degree angle.

🎯 Exam Tip: To define a rectangle, you need to state that it's a quadrilateral with opposite sides equal and all angles right angles, or that it's a parallelogram with one right angle.

Question 2. (iii) What can we say about its lengths of sides, angles and diagonals? Discuss and list them out.
Answer: For this rectangle:
1. Lengths of sides: Opposite sides are equal in length.
2. Angles: All angles are equal and each measures \( 90^\circ \).
3. Diagonals: The diagonals are equal in length and bisect each other (cut each other in half).
These characteristics are fundamental properties that define a rectangle and differentiate it from a general parallelogram. The right angles ensure diagonal equality.
In simple words: In a rectangle, the sides facing each other are equal, all four corners are 90 degrees, and the diagonal lines from corner to corner are also equal and cut each other exactly in the middle.

🎯 Exam Tip: Clearly listing properties for sides, angles, and diagonals is essential for describing any quadrilateral completely.

Question 3. Repeat the above activity, this time with a pair of 45° - 45° - 90° set-squares.
[Image of set-squares forming a square]
(i) How does the figure change now? Is it a parallelogram? It becomes a square! (How did it happen?)
Answer: When using two identical 45°-45°-90° set-squares, the figure formed is a square. It is a parallelogram, and more specifically, a square. This happens because the 45°-45°-90° triangles have two equal sides (the legs), which ensures that when they form a rectangle, all four sides of that rectangle become equal, thus making it a square. This arrangement leads to all sides being equal.
In simple words: When we use two identical special triangles with two 45-degree angles and one 90-degree angle, the shape we make becomes a square because all its sides end up being the same length. Yes, it is still a parallelogram.

🎯 Exam Tip: A square is a special type of rectangle where all four sides are equal, and it is also a special type of rhombus because its diagonals are equal and bisect each other at right angles.

Question 3. (ii) What can we say about its lengths of sides, angles and diagonals? Discuss and list them out.
Answer: For this square:
1. Lengths of sides: All sides are equal in length.
2. Angles: All angles are equal and each measures \( 90^\circ \).
3. Diagonals: The diagonals are equal in length, bisect each other perpendicularly, and also bisect the angles of the square.
These combined properties are unique to a square, which is a highly symmetric quadrilateral. It is both a rectangle and a rhombus.
In simple words: In a square, all its sides are the same length, all its corners are 90 degrees, and the diagonals are equal, cut each other in half, and cross each other at 90-degree angles.

🎯 Exam Tip: A square combines the properties of a rectangle (equal angles, equal diagonals) and a rhombus (equal sides, perpendicular diagonals).

Question 3. (iii) How does it differ from the list we prepared for the rectangle?
Answer: The square differs from a general rectangle in the following ways:
1. All sides are equal: In a square, all four sides are equal, whereas in a rectangle, only opposite sides are equal.
2. Diagonals bisect each other perpendicularly: In a square, the diagonals not only bisect each other but also intersect at a \( 90^\circ \) angle. In a general rectangle, diagonals only bisect each other (not necessarily perpendicularly).
These additional properties make a square a more specific and symmetrical shape than a rectangle. This is why a square is often called a 'special rectangle'.
In simple words: The square is different from a normal rectangle because all its sides are the same length, and its diagonals cut each other at a perfect right angle.

🎯 Exam Tip: When comparing quadrilaterals, focus on the unique properties each shape has in addition to the properties it shares with broader categories.

Question 4. We again use four Identical 30° - 60° - 90° set- squares for this activity. Note carefully how they are placed touching one another.
[Image of four set-squares forming a larger figure]
(j) Do we get a parallelogram now?
Answer: Yes, we get a parallelogram now. When four identical 30°-60°-90° set-squares are arranged as shown, the resulting figure is a parallelogram. This happens because the opposite sides of the combined figure will be formed by identical sides of the set-squares, ensuring they are parallel and equal. This type of arrangement often forms a central void or inner shape.
In simple words: Yes, when these four specific triangle rulers are put together like the picture, the new larger shape that forms is a parallelogram because its opposite sides are parallel.

🎯 Exam Tip: Focus on the overall shape formed by the outer boundaries when identifying the resulting figure, especially when multiple identical shapes are combined.

Try These (Text Book Page No. 195 & 196)

Question 1. Say True or False:
(a) A square is a special rectangle.
(b) A square is a parallelogram.
(c) A square is a special rhombus.
(d) A rectangle is a parallelogram
Answer:
(a) True. A square has all the properties of a rectangle, such as four right angles and equal diagonals. This is why it is considered a special type of rectangle.
(b) True. A square also meets all the conditions of a parallelogram, including opposite sides being parallel and equal.
(c) True. A square has all the properties of a rhombus, like all sides being equal and diagonals intersecting at right angles. It is a special rhombus because it also has four right angles.
(d) True. A rectangle has opposite sides that are parallel and equal, which are the main properties of a parallelogram. Therefore, every rectangle is also a parallelogram.
In simple words: A square is like a super-rectangle and a super-rhombus because it has all their qualities. All squares, rectangles, and rhombuses are also types of parallelograms.

🎯 Exam Tip: Remember the hierarchy of quadrilaterals: a square is a special type of both a rectangle and a rhombus, and all three are special types of parallelograms.

Question 2. Name the quadrilaterals
(a) Which have diagonals bisecting each other.
(b) In which the diagonals are perpendicular bisectors of each other.
(c) Which have diagonals of different lengths.
(d) Which have equal diagonals.
(e) Which have parallel opposite sides.
(f) In which opposite angles are equal.
Answer:
(a) Square, rectangle, parallelogram, rhombus. In all these shapes, the diagonals cut each other exactly in half at their meeting point.
(b) Rhombus and square. For these two shapes, the diagonals not only cut each other in half but also cross at a perfect 90-degree angle.
(c) Parallelogram and Rhombus. In these shapes, the lines connecting opposite corners are not the same length.
(d) Rectangle, square. These shapes have diagonals that are exactly the same length.
(e) Square, Rectangle, Rhombus, parallelogram. All these shapes have at least one pair of opposite sides that run perfectly side-by-side without ever meeting.
(f) Square, rectangle, rhombus, parallelogram. In these shapes, the angles directly across from each other are always the same size.
In simple words: Different four-sided shapes have different rules for their diagonals and sides. Knowing these rules helps you tell them apart.

🎯 Exam Tip: When listing quadrilaterals, ensure you include all types that satisfy the given property. For example, a square satisfies many properties, so it often appears in multiple lists.

Question 3. Two sticks are placed on a ruled sheet as shown. What figure is formed if the four corners of the sticks are joined?
(a) Two unequal sticks. Placed such that their midpoints coincide.
Answer: parallelogram
In simple words: When two sticks of different lengths cross each other exactly in the middle, and you connect their ends, you get a parallelogram. This means its opposite sides are parallel.

🎯 Exam Tip: Remember that in a parallelogram, diagonals bisect each other, but are generally not equal in length and do not necessarily intersect at right angles.

Question 3. (b) Two equal sticks. Placed such that their midpoints coincide.
Answer: Rectangle
In simple words: If two sticks of the same length cross each other exactly in the middle, the shape formed by joining their ends will be a rectangle. This means it has four right angles.

🎯 Exam Tip: For a rectangle, the diagonals are equal and bisect each other. This is a key property that distinguishes it from a general parallelogram.

Question 3. (c) Two unequal sticks. Placed intersecting at mid points perpendicularly.
Answer: Rhombus
In simple words: When two sticks of different lengths cross each other exactly in the middle and also form a right angle, the shape made by joining their ends is a rhombus. This means all its sides are equal in length.

🎯 Exam Tip: The key characteristic of a rhombus is that its diagonals are perpendicular bisectors of each other. This is true even if the diagonals are not equal in length.

Question 3. (d) Two equal sticks. Placed intersecting at mid points perpendicularly.
Answer: Square
In simple words: When two sticks of the same length cross each other exactly in the middle and at a right angle, the shape formed by connecting their ends is a square. This means all its sides are equal and all its angles are right angles.

🎯 Exam Tip: A square is the only quadrilateral whose diagonals are both equal in length and perpendicular bisectors of each other.

Question 3. (e) Two unequal sticks. Tops are not on the same ruling. Bottoms are not on the same ruling. Not cutting at the mid point of either.
Answer: Quadrilateral
In simple words: If two sticks cross each other in a way that is not neat (unequal, not bisecting, and not forming a clear pattern), the shape formed by joining their ends will just be a general four-sided figure, which we call a quadrilateral. It does not have any special properties like parallel sides or equal angles.

🎯 Exam Tip: A quadrilateral is a polygon with four sides. It is the most general type of four-sided figure, with no specific requirements for its sides or angles other than having four of each.

Question 3. (f) Two unequal sticks. Tops on the same ruling. Bottoms on the same ruling. Not necessarily cutting at the mid point of either.
Answer: Trapezium
In simple words: When two sticks are placed such that their top ends lie on one straight line and their bottom ends lie on another straight line (making two parallel lines), the shape formed by connecting their ends is a trapezium. This means it has exactly one pair of parallel sides.

🎯 Exam Tip: For a trapezium, only one pair of opposite sides must be parallel. The diagonals of a trapezium typically do not bisect each other and are not equal unless it is an isosceles trapezium.