Samacheer Kalvi Class 8 Maths Solutions Chapter 5 Geometry Exercise 5.5

I. Construct The Following Parallelograms With The Given Measurements And Find Their Area.

Question 1. ARTS, AR = 6cm, RT = 5cm and \( \angle ART = 70^\circ \).
Answer:
Given: In the Parallelogram ARTS, \( AR = 6 \text{ cm} \), \( RT = 5 \text{ cm} \), and \( \angle ART = 70^\circ \).

**Construction Steps:**
1. Draw a line segment \( AR = 6 \text{ cm} \).
2. At point R on AR, make an angle \( \angle ART = 70^\circ \).
3. With R as the center, draw an arc with a radius of \( 5 \text{ cm} \) that cuts the ray RX (from R at 70 degrees) at point T.
4. Draw a line TY that is parallel to AR and passes through T.
5. With T as the center, draw an arc with a radius of \( 6 \text{ cm} \) that cuts the line TY at point S.
6. Join points A and S.
7. The parallelogram ARTS is now constructed.
**Calculation of area:**
Area of the parallelogram ARTS \( = b \times h \) sq. units
\( = 6 \times 4.7 \)
\( = 28.2 \text{ sq.cm} \)
In simple words: First, draw the base line and the angle given. Then, use a compass to mark the lengths of the other sides and draw parallel lines to complete the shape. Finally, multiply the base by the height to find how much space the parallelogram covers.

🎯 Exam Tip: Always start with a clear rough diagram to visualize the construction. Use a protractor for angles and a ruler for lengths accurately.

Question 2. CAMP, CA = 6cm, AP = 8cm and CP = 5.5cm.
Answer:
Given: In the parallelogram CAMP, \( CA = 6 \text{ cm} \), \( AP = 8 \text{ cm} \), and \( CP = 5.5 \text{ cm} \).

**Construction Steps:**
1. Draw a line segment \( CA = 6 \text{ cm} \).
2. With C as the center, draw an arc with a length of \( 5.5 \text{ cm} \).
3. With A as the center, draw an arc with a length of \( 8 \text{ cm} \).
4. Mark the intersecting point of these two arcs as P.
5. Draw a line PX parallel to CA through P.
6. With P as the center, draw an arc with a radius of \( 6 \text{ cm} \) that cuts PX at M.
7. Join AM.
8. CAMP is the required parallelogram.
**Calculation of area:**
Area of the Parallelogram CAMP \( = b \times h \) sq. units
\( = 6 \times 5.5 \)
\( = 33 \text{ sq.cm} \)
In simple words: Start by drawing one side of the parallelogram. Then, use the given lengths of the diagonals or other sides to find the third point. Complete the parallelogram by drawing parallel lines. Finally, calculate its area by multiplying the base by its vertical height.

🎯 Exam Tip: When constructing a parallelogram using sides and a diagonal, constructing one of the triangles first (e.g., \( \triangle CAP \)) is a key step.

Question 3. EARN, ER = 10cm, AN = 7cm and \( \angle EOA = 110^\circ \) where \( \overline{\mathrm{ER}} \) and \( \overline{\mathrm{AN}} \) intersect at O.
Answer:
Given: In the parallelogram EARN, \( ER = 10 \text{ cm} \), \( AN = 7 \text{ cm} \), and \( \angle EOA = 110^\circ \). The diagonals \( \overline{\mathrm{ER}} \) and \( \overline{\mathrm{AN}} \) intersect at O.

**Construction Steps:**
1. Draw a line segment PX. Mark a point O on PX.
2. At point O on PX, make an angle \( \angle EOA = 110^\circ \).
3. With O as the center, draw arcs with a radius of \( 3.5 \text{ cm} \) on either side of PX, cutting the line YZ (representing the angle line) at A and N. This is because diagonals bisect each other, so \( AO = ON = 7/2 = 3.5 \text{ cm} \).
4. With O as the center, draw arcs with a radius of \( 5 \text{ cm} \) along the line ER, cutting at E and R. This is because diagonals bisect each other, so \( EO = OR = 10/2 = 5 \text{ cm} \).
5. Join AE, ER, RN, and NA.
6. EARN is the required parallelogram.
**Calculation of area:**
Area of the Parallelogram EARN \( = b \times h \) sq. units
\( = 10 \times 5.5 \)
\( = 55 \text{ sq.cm} \)
In simple words: When given diagonals and the angle between them, first draw the diagonals intersecting at the given angle, making sure they bisect each other. Then, connect the endpoints of the diagonals to form the parallelogram. Find the area by multiplying the base by its corresponding height.

🎯 Exam Tip: Remember that the diagonals of a parallelogram bisect each other. This means you use half the length of each diagonal when marking points from the intersection.

Question 4. GAIN, GA = 7.5cm, GI = 9cm and \( \angle GAI = 100^\circ \).
Answer:
Given: In the parallelogram GAIN, \( GA = 7.5 \text{ cm} \), \( GI = 9 \text{ cm} \), and \( \angle GAI = 100^\circ \).

**Construction Steps:**
1. Draw a line segment \( GA = 7.5 \text{ cm} \).
2. At point A, make an angle \( \angle GAI = 100^\circ \).
3. With G as the center, draw an arc with a radius of \( 9 \text{ cm} \) that cuts the ray AX at I. Then, join GI.
4. Draw a line IY parallel to GA through I.
5. With I as the center, draw an arc with a radius of \( 7.5 \text{ cm} \) on IY that cuts at N. Then, join GN.
6. GAIN is the required parallelogram.
**Calculation of area:**
Area of the Parallelogram GAIN \( = b \times h \) sq. units
\( = 7.5 \times 3.9 \)
\( = 29.25 \text{ sq. cm} \)
In simple words: First, draw the base and the given angle at one end. Then, use a compass to mark the length of the adjacent side. After that, draw parallel lines and connect the remaining points to complete the parallelogram. Finally, calculate the area using the base and the measured perpendicular height.

🎯 Exam Tip: When given two sides and an included angle, ensure the angle is correctly drawn at the vertex where the two given sides meet.

II. Construct The Following Rhombuses With The Given Measurements And Also Find Their Area.

Question (i) FACE, FA = 6 cm and FC = 8 cm
Answer:
Given: For the rhombus FACE, side \( FA = 6 \text{ cm} \) and diagonal \( FC = 8 \text{ cm} \).

**Construction Steps:**
1. Draw a line segment \( FA = 6 \text{ cm} \).
2. With F and A as centers, draw arcs with radii of \( 8 \text{ cm} \) and \( 6 \text{ cm} \) respectively. Let them cut at C.
3. Join FC and AC.
4. With F and C as centers, draw arcs with a radius of \( 6 \text{ cm} \) each. Let them cut at E.
5. Join FE and EC.
6. FACE is the required rhombus.
**Calculation of Area:**
Area of the rhombus \( = \frac { 1 }{ 2 } \times d_1 \times d_2 \) sq. units
\( = \frac { 1 }{ 2 } \times 8 \times 9 \)
\( = 36 \text{ cm}^2 \)
In simple words: A rhombus has four equal sides. Start by drawing one side. Then, use the given diagonal length to find a third point, forming a triangle. Complete the rhombus by finding the fourth point using the side length and connecting all vertices. Calculate its area by multiplying half of its two diagonals.

🎯 Exam Tip: For rhombuses, remember that all four sides are equal, and the diagonals bisect each other at right angles. This is useful for both construction and area calculation.

Question (ii) CAKE, CA = 5 cm and \( \angle A = 65^\circ \)
Answer:
Given: For the rhombus CAKE, side \( CA = 5 \text{ cm} \) and angle \( \angle A = 65^\circ \).

**Construction Steps:**
1. Draw a line segment \( CA = 5 \text{ cm} \).
2. At point A on AC, make an angle \( \angle CAX = 65^\circ \).
3. With A as the center, draw an arc with a radius of \( 5 \text{ cm} \). Let it cut AX at K.
4. With K and C as centers, draw arcs with a radius of \( 5 \text{ cm} \) each. Let them cut at E.
5. Join KE and CE.
6. CAKE is the required rhombus.
**Calculation of Area:**
Area of the rhombus \( = \frac { 1 }{ 2 } \times d_1 \times d_2 \) sq. units
\( = \frac { 1 }{ 2 } \times 5.4 \times 8.5 \)
\( = 22.95 \text{ cm}^2 \)
In simple words: Since all sides of a rhombus are equal, draw the first side and the given angle. Then, use the side length to find the next two vertices. Connect the points to complete the shape. To find its area, multiply half of the lengths of its two diagonals.

🎯 Exam Tip: When given an angle, remember that opposite angles in a rhombus are equal. This can help verify your construction.

Question (iii) LUCK, LC = 7.8 cm and UK = 6 cm
Answer:
Given: For the rhombus LUCK, diagonal \( LC = 7.8 \text{ cm} \) and diagonal \( UK = 6 \text{ cm} \).

**Construction Steps:**
1. Draw a line segment \( LC = 7.8 \text{ cm} \).
2. Draw the perpendicular bisector XY of LC. Let it cut LC at 'O'.
3. With O as the center, draw an arc of radius \( 3 \text{ cm} \) on either side of O (along XY), cutting OX at K and OY at U. (Since diagonals bisect each other, \( \frac{UK}{2} = \frac{6}{2} = 3 \text{ cm} \)).
4. Join LU, UC, CK, and LK.
5. LUCK is the required rhombus.
**Calculation of Area:**
Area of the rhombus \( = \frac { 1 }{ 2 } \times d_1 \times d_2 \) sq. units
\( = \frac { 1 }{ 2 } \times 7.8 \times 6 \text{ cm}^2 \)
\( = 23.4 \text{ cm}^2 \)
In simple words: When given both diagonals of a rhombus, first draw one diagonal. Then, draw its perpendicular bisector. Mark the half-length of the second diagonal on the bisector to find the other two vertices. Connect all points to complete the rhombus. Its area is half the product of its diagonals.

🎯 Exam Tip: Always bisect the diagonals perpendicularly when constructing a rhombus using diagonal lengths. This is a fundamental property of rhombuses.

Question (iv) PARK, PR = 9 cm and \( \angle P = 70^\circ \)
Answer:
Given: For the rhombus PARK, diagonal \( PR = 9 \text{ cm} \) and angle \( \angle P = 70^\circ \).

**Construction Steps:**
1. Draw a line segment \( PR = 9 \text{ cm} \).
2. At P, make angles of \( \angle RPX = 35^\circ \) and \( \angle RPY = 35^\circ \) on either side of PR. (Since diagonals bisect the angles, \( 70^\circ / 2 = 35^\circ \)).
3. At R, make angles of \( \angle PRQ = 35^\circ \) and \( \angle PRS = 35^\circ \) on either side of PR.
4. Let PX and RQ cut at A, and PY and RS cut at K.
5. PARK is the required rhombus.
**Calculation of Area:**
Area of the rhombus \( = \frac { 1 }{ 2 } \times d_1 \times d_2 \) sq. units
\( = \frac { 1 }{ 2 } \times 9 \times 6.2 \text{ cm}^2 \)
\( = 27.9 \text{ cm}^2 \)
In simple words: For a rhombus where you know one diagonal and an angle, draw the diagonal first. Then, at the ends of this diagonal, draw angles that are half of the given angle on both sides. Where these new lines cross, you will find the other two vertices of the rhombus. Connect all the points. Calculate the area by multiplying half of its two diagonals.

🎯 Exam Tip: Remember that the diagonals of a rhombus bisect the angles at the vertices. So, if an angle is \( X^\circ \), construct with angles of \( X/2^\circ \).

III. Construct The Following Rectangles With The Given Measurements And Also Find Their Area.

Question (i) HAND, HA = 7cm and AN = 4 cm
Answer:
Given: For the rectangle HAND, length \( HA = 7 \text{ cm} \) and width \( AN = 4 \text{ cm} \).

**Construction Steps:**
1. Draw a line segment \( HA = 7 \text{ cm} \).
2. At H, construct a perpendicular line HX to HA.
3. With H as the center, draw an arc of radius \( 4 \text{ cm} \) that cuts HX at D.
4. With A and D as centers, draw arcs of radii \( 4 \text{ cm} \) and \( 7 \text{ cm} \) respectively. Let them cut at N.
5. Join AN and DN.
6. HAND is the required rectangle.
**Calculation of area:**
Area of the rectangle HAND \( = l \times b \) sq. units
\( = 7 \times 4 \text{ cm}^2 \)
\( = 28 \text{ cm}^2 \)
In simple words: A rectangle has opposite sides equal and all angles are 90 degrees. Start by drawing the length, then draw a perpendicular line. Mark the width on the perpendicular line. Complete the rectangle by drawing parallel lines. Calculate its area by multiplying its length by its width.

🎯 Exam Tip: Always construct a 90-degree angle accurately at the corners of a rectangle to ensure it's truly rectangular.

Question (ii) LAND, LA = 8cm and AD = 10 cm
Answer:
Given: For the rectangle LAND, length \( LA = 8 \text{ cm} \) and diagonal \( AD = 10 \text{ cm} \).

**Construction Steps:**
1. Draw a line segment \( LA = 8 \text{ cm} \).
2. At L, construct a perpendicular line LX to LA.
3. With A as the center, draw an arc of radius \( 10 \text{ cm} \) that cuts LX at D.
4. With A as the center and LD as radius, draw an arc. Also, with D as the center and LA as radius, draw another arc. Let them cut at N.
5. Join DN and AN.
6. LAND is the required rectangle.
**Calculation of area:**
Area of the rectangle LAND \( = l \times b \) sq. units
\( = 8 \times 5.8 \text{ cm}^2 \)
\( = 46.4 \text{ cm}^2 \)
In simple words: Start by drawing the length of the rectangle. Then, use the diagonal length to find the width by making a right-angled triangle. Once you have the length and width, complete the rectangle by drawing parallel lines. Finally, multiply the length by the width to get the area.

🎯 Exam Tip: When given a diagonal and a side, use the Pythagorean theorem concept to find the missing side (width) if not provided directly, or construct it using compass arcs.

IV. Construct The Following Squares With The Given Measurements And Also Find Their Area.

(i) EAST, EA = 6.5 cm
Answer:To construct the square EAST with a side length EA = 6.5 cm and find its area:

**Steps for Construction:**
1. Draw a line segment EA that measures 6.5 cm.
2. At point E, construct a ray EX perpendicular to EA, forming a 90-degree angle.
3. Using E as the center, draw an arc with a radius of 6.5 cm. This arc should intersect the ray EX at point T.
4. Now, using A as the center, draw an arc with a radius of 6.5 cm.
5. Then, using T as the center, draw another arc with a radius of 6.5 cm. These two arcs should meet at point S.
6. Join points A to S and T to S.
7. The figure EAST is the required square. A square has four equal sides and four right angles.

**Calculation of Area:**
The formula for the area of a square is side \( \times \) side, or side squared.
Given side \( (a) = 6.5 \) cm.
Area \( = a^2 = (6.5 \, \text{cm})^2 = 42.25 \, \text{cm}^2 \).

In simple words: First, draw a line for one side of the square. Then make 90-degree angles at the ends and measure out the other sides with your compass. Connect the last points to complete the square, then multiply the side length by itself to get the area.

🎯 Exam Tip: When constructing a square with a given side, ensure all sides are equal and all angles are 90 degrees. Mark arcs carefully to locate the final vertices.

(ii) WEST, WS = 7.5 cm
Answer:To construct the square WEST with a diagonal WS = 7.5 cm and find its area:

**Steps for Construction:**
1. Draw a line segment WS that measures 7.5 cm.
2. Construct the perpendicular bisector XY for the line segment WS. This bisector will cut WS at its midpoint, let's call it O.
3. Using O as the center, draw an arc with a radius of 3.75 cm (half of the diagonal length) along both sides of the bisector XY. These arcs will cut the line OX at point T and OY at point E.
4. Join the points W to T, T to S, S to E, and E to W.
5. The figure WEST is the required square. Diagonals of a square bisect each other at right angles.

**Calculation of Area:**
The area of a square can be found using its diagonal length \( d \) with the formula: Area \( = \frac{1}{2} d^2 \).
Given diagonal \( (d) = 7.5 \) cm.
Area \( = \frac{1}{2} \times (7.5 \, \text{cm})^2 \)
\( = \frac{1}{2} \times 56.25 \, \text{cm}^2 \)
\( = 28.125 \, \text{cm}^2 \).
Alternatively, the side length \( a = \frac{\text{diagonal}}{\sqrt{2}} = \frac{7.5}{\sqrt{2}} \approx 5.303 \, \text{cm} \).
So, Area \( = a^2 \approx (5.3 \, \text{cm}) \times (5.3 \, \text{cm}) = 28.09 \, \text{cm}^2 \).

In simple words: Start by drawing the given diagonal. Then, find its exact middle and draw a straight line through it, going both ways at a right angle. Measure half the diagonal length from the middle point along this new line to find the other two corners of the square. Connect all four corners to finish the square. The area can be found using half of the diagonal squared.

🎯 Exam Tip: When constructing a square from its diagonal, always draw the diagonal first, then construct its perpendicular bisector to find the other two vertices accurately. Remember that diagonals are equal and bisect each other at 90 degrees.