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## Solved Problems on Different Types of Quadrilaterals

1. Construct a parallelogram PQRS in which PQ = 5.2 cm, QR = 4.7 cm and PR = 7.6 cm.

## Solution:

Steps of Construction:

Given that a parallelogram PQRS in which PQ = 5.2 cm, QR = 4.7 cm and PR = 7.6 cm.

1. Draw a line segment of length 5.2 cm and mark the ends as P and Q.

2. Take the point P as a center and draw an arc by taking the radius 7.6 cm.

3. Next, take point Q as a center and draw an arc by taking the radius 4.7 cm. Mark the point as R where the two arcs cross each other. Join the points Q and R as well as P and R.

Note: A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

4. By taking the point P as a center, draw an arc with a radius of 4.7 cm.

5. By taking the point R as a center, draw an arc with a radius of 5.2 cm.

6. Mark the point as S where the two arcs cross each other. Join the points R and S as well as P and S.

PQRS is a required parallelogram.

2. Construct a parallelogram PQRS in which PQ = 4.3 cm, PS = 4 cm and QS = 6.8 cm.

## Solution:

Steps of Construction:

Given that a parallelogram PQRS in which PQ = 4.3 cm, PS = 4 cm and QS = 6.8 cm.

1. Draw a line segment of length 4.3 cm and mark the ends as P and Q.

2. Take the point P as a center and draw an arc by taking the radius 4 cm.

3. Next, take point Q as a center and draw an arc by taking the radius 6.8 cm. Mark the point as S where the two arcs cross each other. Join the points Q and S as well as P and S.

Note: A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

4. By taking the point Q as a center, draw an arc with a radius of 4 cm.

5. By taking the point S as a center, draw an arc with a radius of 4.3 cm.

6. Mark the point as R where the two arcs cross each other. Join the points R and S as well as R and Q.

PQRS is a required parallelogram.

3. Construct a parallelogram ABCD in which BC = 6 cm, AB = 4 cm and ∠ABC = 60°.

## Solution:

Steps of Construction:

Given that a parallelogram ABCD in which BC = 6 cm, AB = 4 cm and ∠ABC = 60°.

1. Draw a line segment of length 4 cm and mark the ends as A and B.

2. Take point B as a center and make a point by taking 60º using a protector.

3. Next, take point B as a center and draw an arc by taking the radius 6 cm. Mark the point as C where the point and arc cross each other. Join the points C and B.

Note: A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

4. By taking the point C as a center, draw an arc with a radius of 4 cm.

5. By taking point A as a center, draw an arc with a radius of 6 cm.

6. Mark the point as D where the two arcs cross each other. Join the points D and C as well as D and A.

ABCD is a required parallelogram.

4. Construct a parallelogram PQRS in which QR = 5 cm, ∠PQR = 120° and RS = 4.8 cm.

## Solution:

Steps of Construction:

Given that a parallelogram PQRS in which QR = 5 cm, ∠PQR = 120° and RS = 4.8 cm.

1. Draw a line segment of length 5 cm and mark the ends as Q and R.

2. Take point R as a center and make a point by taking 120º using a protector.

3. Next, take point R as a center and draw an arc by taking the radius 4.8 cm. Mark the point as S where the point and arc cross each other. Join the points S and R.

Note: A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

4. By taking the point S as a center, draw an arc with a radius of 5 cm.

5. By taking point Q as a center, draw an arc with a radius of 4.8 cm.

6. Mark the point as P where the two arcs cross each other. Join the points P and S as well as P and Q.

PQRS is a required parallelogram.

5. Construct a parallelogram PQRS, one of whose sides is 4.4 cm and whose diagonals are 5.6 cm and 7 cm. Measure the other side?

## Solution:

Steps of Construction:

Given that a parallelogram PQRS, one of whose sides is 4.4 cm and whose diagonals are 5.6 cm and 7 cm.

1. Draw a line segment of length 4.4 cm and mark the ends as P and Q.

2. Make the diagonals half to get the exact vertices of a parallelogram. Take point P as a center and draw an arc by taking the radius 2.8 cm.

3. Next, take point Q as a center and draw an arc by taking the radius 3.5 cm. Mark the point as O where the point and arc cross each other. Join the points P and O, Q and O.

4. Extend the line PO with a radius of 3.5 cm and mark it as R.

5. Extend the line QO with a radius of 2.8 cm and mark it as S.

6. Join the points P and S as well as Q and R, R and S.

PQRS is a required parallelogram.

6. Construct a parallelogram PQRS in which PQ = 6.5 cm, PR = 3.4 cm, and the altitude PL from P is 2.5 cm. Draw the altitude from R and measure it?

## Solution:

Steps of Construction:

Given that a parallelogram PQRS in which PQ = 6.5 cm, PR = 3.4 cm, and the altitude PL from P is 2.5 cm.

1. Draw a line segment of length 6.5 cm and mark the ends as P and Q.

2. Take point P as a center and draw a perpendicular line PX by taking the radius 2.5 cm and mention that point as L. Draw a parallel line to PQ at a point L.

3. Next, take point P as a center and draw an arc by taking the radius of 3.4 cm. Mark the point as R where the point and arc cross each other. Join the points P and R, Q and R.

4. Take point R as a center and draw an arc by taking the radius of 6.5 cm. Mark the point as S where the point and arc cross each other. Join the points R and S, P and S.

PQRS is a required parallelogram.

7. Construct a parallelogram PQRS, in which diagonal PR = 3.8 cm, diagonal QS = 4.6 cm, and the angle between PR and QS is 60°.

## Solution:

Steps of Construction:

Given that a parallelogram PQRS, in which diagonal PR = 3.8 cm, diagonal QS = 4.6 cm, and the angle between PR and QS is 60°.

1. Draw a line segment of length 3.8 cm and mark the ends as P and R.

2. Bisect the line PR and point it as O.

3. Next, take point O and make a point by taking the angle 60° by taking the O as a center.

4. Take point O as a center and draw an arc by taking the radius of 2.3 cm on both sides of O and name them as Q and S. Join the points PQ, QR, RS, SP.

PQRS is a required parallelogram.

8. Construct a rectangle PQRS whose adjacent sides are 11 cm and. 8.5 cm.

## Solution:

Steps of Construction:

Given that a rectangle PQRS whose adjacent sides are 11 cm and. 8.5 cm.

1. Draw a line segment of length 11 cm and mark the ends as P and Q.

2. Take point P as a center and make a point by taking 90º using a protector and make the point as E.

Rectangle: All angles of a rectangle are 90º.

3. Next, take point P as a center and draw an arc by taking the radius 8.5 cm. Mark the point as S where the point and arc cross each other. Join the points S and P.

4. Take point Q as a center and draw an arc by taking the radius of 8.5 cm.

5. Also, take point S as a center and draw an arc by taking the radius of 11 cm.

Mark the point as R where the two arcs cross each other. Join the points S and R, Q and R.

PQRS is a required rectangle.

9. Construct a square, each of whose sides measures 5.4 cm.

## Solution:

Steps of Construction:

Given that a square, each of whose sides measures 5.4 cm.

1. Draw a line segment of length 5.4 cm and mark the ends as P and Q.

2. Take point P as a center and make a point by taking 90º using a protector and make the point as E.

Square: All angles of a square are 90º.

3. Next, take point P as a center and draw an arc by taking the radius 5.4 cm. Mark the point as S where the point and arc cross each other. Join the points S and P.

4. Take point Q as a center and make a point by taking 90º using a protector. Take point Q as a center and draw an arc by taking the radius of 5.4 cm.

5. Also, take point S as a center and draw an arc by taking the radius of 5.4 cm. Mark the point as R where the two arcs cross each other. Join the points S and R, Q and R.

PQRS is a required square.

10. Construct a square, each of whose diagonals measures 5.6 cm.

## Solution:

Steps of Construction:

Given that a square, each of whose diagonals measures 5.6 cm.

1. Draw a line segment of length 5.6 cm and mark the ends as P and R.

2. Bisect the line PR and take half of its radius 2.8 cm. Take point P as a center and draw an arc by taking the radius 2.8 cm.

3. Next, take point R as a center and draw an arc by taking the radius 2.8 cm. Mark the point as S where the point and arc cross each other. Join the points S and P, S and R.

4. Similarly, draw two arcs with 2.8 cms and make the point as Q.

5. Join the points P and Q, S and Q.

PQRS is a required square.

11. Construct a rectangle ABCD in which BD = 3.6 cm and diagonal AD = 6 cm. Measure the other side of the rectangle.

## Solution:

Steps of Construction:

Given that a rectangle ABCD in which BD = 3.6 cm and diagonal AD = 6 cm.

1. Draw a line segment of length 3.6 cm and mark the ends as B and D.

2. Take point B as a center and make a point by taking 90º using a protector and make the point as E.

Rectangle: All angles of a rectangle are 90º.

3. Next, take point C as a center and draw an arc by taking the radius 6 cm. Mark the point as A where the point and arc cross each other. Join the points A and B, A and D.

4. Similarly, draw two arcs and make the point as C.

5. Join the points A and C, D and C.

ABCD is a required rectangle.

12. Construct a rhombus PQRS when the length measures of the diagonals are 8 cm and 6 cm.

## Solution:

Steps of Construction:

Given that a rhombus PQRS when the length measures of the diagonals are 8 cm and 6 cm.

1. Draw a line segment of length 8 cm and mark the ends as P and R.

2. Draw perpendicular bisector XY of PR meeting PR at O.

3. Next, From O cut off OS = 1/2 × 6 cm = 3 cm along OX and OQ = 1/2 × 6 cm =3 cm along OY.

4. Join PQ, QR, RS, and SP.

PQRS is a required rhombus.

13. Construct a rhombus PQRS in which PQ = 4 cm and diagonal PR is 6.5 cm.

## Solution:

Steps of Construction:

Given that a rhombus PQRS in which PQ = 4 cm and diagonal PR is 6.5 cm.

1. Draw a line segment of length 4 cm and mark the ends as P and Q.

2. Take the point Q as a center and draw an arc by taking the radius 4 cm.

3. Take the point P as a center and draw an arc by taking the radius 6.5 cm and name it as R.

4. Join PR and QR.

5. Take the point R as a center and draw an arc by taking the radius 4 cm.

6. Take the point P as a center and draw an arc by taking the radius 4 cm. Mark the point as S where the point and arc cross each other. Join the points S and P, S and R.

PQRS is a required rhombus.

14. Draw a rhombus ABCD whose side is 7.2 cm and one angle is 60°.

## Solution:

Steps of Construction:

Given that a rhombus ABCD whose side is 7.2 cm and one angle is 60°.

1. Draw a line segment of length 7.2 cm and mark the ends as A and B.

2. Take point A as a center and make a point by taking 60º using a protector.

3. Next, take point A as a center and draw an arc by taking the radius 7.2 cm. Mark the point as D where the point and arc cross each other. Join the points D and A.

4. Take point D as a center and draw an arc by taking the radius 7.2 cm. Take point B as a center and draw an arc by taking the radius of 7.2 cm.

5. Mark the point as C where the two arcs cross each other. Join the points C and B, C and D.

ABCD is a required rhombus.

15. Construct a trapezium PQRS in which PQ = 6 cm, QR = 4 cm, RS = 3.2 cm, ∠Q = 75° and SR ∥ PQ.

## Solution:

Steps of Construction:

Given that a trapezium PQRS in which PQ = 6 cm, QR = 4 cm, RS = 3.2 cm, ∠Q = 75° and SR ∥ PQ.

1. Draw a line segment of length 6 cm and mark the ends as P and Q.

2. Take point Q as a center and make a point by taking 75º using a protector.

3. Next, take point Q as a center and draw an arc by taking the radius 4 cm. Mark the point as R where the point and arc cross each other. Join the points Q and R.

4. RS || PQ, so angle Q + angle R = 180º, angle C = 105º as they are interior angles.

5. Take point R as a center and make a point by taking 105º using a protector.

6. Next, take point R as a center and draw an arc by taking the radius 3.2 cm. Mark the point as S where the point and arc cross each other. Join the points R and S, S and P.

PQRS is a required rhombus.

16. Draw a trapezium PQRS in which PQ ∥ SR, PQ = 7 cm, QR = 5 cm, PS = 6.5 cm and ∠Q = 60°.

## Solution:

Steps of Construction:

Given that a trapezium PQRS in which PQ ∥ SR, PQ = 7 cm, QR = 5 cm, PS = 6.5 cm and ∠Q = 60°.

1. Draw a line segment of length 7 cm and mark the ends as P and Q.

2. Take point Q as a center and make a point by taking 60º using a protector.

3. Next, take point Q as a center and draw an arc by taking the radius 5 cm. Mark the point as R where the point and arc cross each other. Join the points Q and R.

4. RS || PQ, so angle Q + angle R = 180º, angle C = 120º as they are interior angles.

5. Take point R as a center and make a point by taking 120º using a protector.

6. Next, take point R as a center and draw an arc by taking the radius 6.5 cm. Mark the point as S where the point and arc cross each other. Join the points R and S, S and P.

PQRS is a required rhombus.