RBSE Solutions Class 9 Maths Chapter 9 Quadrilaterals Exercise 9.3

Question 1. Construct a quadrilateral ABCD, when AB = 3.5 cm, BC = 4.8 cm, CD = 5.1 cm, AD = 4.4 cm and diagonal AC = 5.9 cm.
Answer:

1. First, draw a line segment AB that is 3.5 cm long. This will be the base of your quadrilateral.
2. Using point A as the center, draw an arc with a compass set to 5.9 cm.
3. Now, using point B as the center, draw another arc with a compass set to 4.8 cm. These two arcs will cross each other at a point, which you will label C.
4. Connect point A to point C and point B to point C with straight lines. These lines form two sides and one diagonal of the quadrilateral.
5. Next, use point C as the center and draw an arc with a radius of 5.1 cm. Then, from point A, draw another arc with a radius of 4.4 cm. These new arcs will meet at a point; label this point D.
6. Finally, connect point A to point D and point D to point C. You have now successfully drawn the quadrilateral ABCD as required.

In simple words: Start with the base AB. Use a compass to draw arcs from A and B to find point C. Then, use arcs from A and C to find point D. Connect all points to complete the quadrilateral.

🎯 Exam Tip: Always make sure to use sharp pencils and accurate compass settings to ensure the constructed quadrilateral is precise.

Question 2. Construct a quadrilateral PQRS, when PQ = 4 cm, QR = 3 cm, QS = 4.8 cm, PS = 3.5 cm and PR = 5 cm.
Answer:

1. Begin by drawing a line segment PQ, which will be 4 cm long and serve as the base.
2. Place your compass at point P and draw an arc with a radius of 5 cm (for diagonal PR).
3. From point Q, draw another arc with a radius of 3 cm (for side QR). The point where this arc crosses the previous one is R. This step helps establish the first triangle, PQR.
4. Next, place your compass at point P and draw an arc with a radius of 3.5 cm (for side PS). Simultaneously, from point Q, draw another arc with a radius of 4.8 cm (for diagonal QS). These arcs will meet at point S.
5. Connect points Q to R, R to S, S to P. Also, draw the diagonals P to R and S to Q. The figure PQRS is the quadrilateral you needed to construct.

In simple words: Draw the base PQ first. Then, use the given lengths to draw arcs from P and Q to find R. After that, draw arcs from P and Q again to find S. Connect all the points correctly.

🎯 Exam Tip: When constructing a quadrilateral with given sides and diagonals, it's often easiest to form triangles using the diagonals first.

Question 3. Construct a quadrilateral ABCD, when AB = 4 cm, BC = 4.5 cm, CD = 3.5 cm, AD = 3 cm and \( \angle A = 60^\circ \).
Answer:

1. First, draw a line segment AB that is 4 cm long as the base. This is the starting point for your construction.
2. At point A, use a protractor to draw an angle of 60 degrees. Extend this line as a ray.
3. With point A as the center, draw an arc with a radius of 3 cm along the 60-degree line you just drew. This will mark point D (since AD is 3 cm).
4. Next, from point D, draw an arc with a radius of 3.5 cm (for side CD). Then, from point B, draw another arc with a radius of 4.5 cm (for side BC). These two arcs will intersect and give you point C.
5. Connect point C to point D and point B to point C.
6. The figure ABCD is the quadrilateral you needed to construct.

In simple words: Start with side AB and draw the 60-degree angle at A. Mark D using the length of AD. Then, use arcs from D and B to find C. Connect the remaining sides.

🎯 Exam Tip: Drawing angles accurately with a protractor is crucial for constructions involving angles. Double-check your angle measurements.

Question 4. Construct a quadrilateral ABCD, when AB = 3.5 cm, BC = 3 cm, AD = 2.5 cm, AC = 4.5 cm and BD = 4 cm.
Answer:

1. Start by drawing a line segment AB, 3.5 cm long, to be your base. Drawing the longest side first can sometimes make the construction easier.
2. Using point A as the center, draw an arc with a radius of 4.5 cm (for diagonal AC).
3. Then, using point B as the center, draw an arc with a radius of 3 cm (for side BC). These arcs will meet to give you point C.
4. Now, from point A, draw an arc with a radius of 2.5 cm (for side AD). At the same time, from point B, draw another arc with a radius of 4 cm (for diagonal BD). These two arcs will intersect at point D.
5. Connect the points to form the quadrilateral: A to C, B to C, A to D, B to D, and C to D.
6. The constructed figure is the desired quadrilateral ABCD.

In simple words: First, draw AB. Then, use compass arcs from A and B to locate point C. After that, use more arcs from A and B to locate point D. Join all points to complete the quadrilateral.

🎯 Exam Tip: When given diagonals, treat them as sides of internal triangles. For example, AC and BC form triangle ABC, which can be constructed first.

Question 5. Construct a quadrilateral PQRS, when PQ = 3 cm, QR = 4 cm, PS = 4.5 cm, QS = 5.5 cm and PR = 6 cm.
Answer:

1. Begin by drawing a line segment PQ, which is 3 cm long and will serve as the base.
2. Place your compass at point P and draw an arc with a radius of 6 cm (for diagonal PR).
3. Then, from point Q, draw another arc with a radius of 4 cm (for side QR). The intersection of these two arcs will give you point R. This completes the triangle PQR.
4. Next, with point P as the center, draw an arc with a radius of 4.5 cm (for side PS). Simultaneously, from point Q, draw another arc with a radius of 5.5 cm (for diagonal QS). These arcs will meet at point S.
5. Connect the points: Q to R, R to S, S to P. Also, draw the diagonals P to R and S to Q.
6. The resulting figure is the required quadrilateral PQRS.

In simple words: Draw PQ first. Find R using lengths PR and QR from P and Q. Then find S using lengths PS and QS from P and Q. Join the points to draw the quadrilateral.

🎯 Exam Tip: Always label the vertices as you find them to avoid confusion during the construction steps.

Question 6. Construct a quadrilateral ABCD, when AB = BC = 3 cm, AD = 5 cm, \( \angle A = 90^\circ \) and \( \angle B = 120^\circ \).
Answer:

1. First, draw a line segment AB that is 3 cm long as the base. This sets the foundation for your drawing.
2. At point A, use a protractor to draw an angle of 90 degrees. Extend this line (ray) from A.
3. At point B, draw a 120-degree angle using a protractor. Extend this line (ray) from B.
4. From point A, measure 5 cm along the 90-degree line and mark point D (since AD is 5 cm). From point B, measure 3 cm along the 120-degree line and mark point C (since BC is 3 cm).
5. Finally, connect point C to point D to complete the quadrilateral.
6. You have now constructed the quadrilateral ABCD as required.

In simple words: Draw AB as the base. At A, make a 90-degree angle and mark D. At B, make a 120-degree angle and mark C. Then, just connect C and D.

🎯 Exam Tip: When angles are given, ensure your protractor is aligned correctly with the base line at the vertex for accurate angle drawing.

Question 7. To construct a quadrilateral ABCD, when AB = 3.8 cm, BC = 2.5 cm, CD = 4.5 cm, \( \angle B = 30^\circ \) and \( \angle C = 150^\circ \).
Answer:

1. Start by drawing a line segment BC, 2.5 cm long, as your base. This helps to set up the two given angles.
2. At point B, draw an angle of 30 degrees. At point C, draw an angle of 150 degrees. Extend these lines as rays.
3. Now, from point B, measure 3.8 cm along the 30-degree line and mark point A (since AB is 3.8 cm). Then, from point C, measure 4.5 cm along the 150-degree line and mark point D (since CD is 4.5 cm).
4. Connect point A to point D with a straight line.
5. The constructed figure is the quadrilateral ABCD you needed to draw.

In simple words: Draw BC first. Then, make angles at B and C. Mark A and D using their given lengths along the angle lines. Finally, connect A and D.

🎯 Exam Tip: Always make sure to use a protractor for precise angle measurements, as even a small error can lead to an inaccurate final figure.

Question 8. Construct a quadrilateral PQRS, when PQ = 3 cm, QR = 3.5 cm, \( \angle Q = 90^\circ \) and \( \angle P = 105^\circ \), \( \angle R = 120^\circ \).
Answer:

1. First, draw a line segment PQ, which is 3 cm long, as the base of the quadrilateral.
2. At point P, draw a ray PX so that the angle \( \angle QPX \) is 105 degrees.
3. At point Q, draw another ray QZ, making an angle of 90 degrees \( (\angle PQZ) \). Along this ray QZ, measure 3.5 cm from Q and mark point R. This creates the side QR.
4. At point R, draw an angle of 120 degrees such that its arm extends to meet the ray PX. The point where they meet is S. This completes the side RS and PS.
5. The resulting figure PQRS is the quadrilateral you needed to construct.

In simple words: Draw PQ. Make angles at P and Q. Mark R along the Q angle line. Then make an angle at R, and where that line meets the P angle line, that is S.

🎯 Exam Tip: When three angles are given, ensuring that your angle measurements are precise will lead to a perfect intersection for the fourth vertex.

Question 9. Construct a quadrilateral PQRS, when PQ = 2.5 cm, QR = 3.7 cm, \( \angle P = 90^\circ \), \( \angle Q = 120^\circ \) and \( \angle R = 90^\circ \).
Answer:

We know that the sum of interior angles of a quadrilateral is \( 360^\circ \). This helps us find any missing angle if others are known.

\( \therefore \angle P + \angle Q + \angle R + \angle S = 360^\circ \)
\( \therefore 90^\circ + 120^\circ + 90^\circ + \angle S = 360^\circ \)
\( 300^\circ + \angle S = 360^\circ \)
\( \implies \angle S = 360^\circ - 300^\circ \)
\( \implies \angle S = 60^\circ \)

1. First, draw a line segment PQ, which is 2.5 cm long, as the base.
2. At point P, draw a ray PX so that the angle \( \angle XPQ \) is 90 degrees.
3. At point Q, draw another ray QY so that the angle \( \angle PQY \) is 120 degrees.
4. Using point Q as the center, draw an arc with a radius of 3.7 cm along the ray QY. This marks point R (since QR is 3.7 cm).
5. At point R, draw an angle of 90 degrees such that its arm extends until it meets the ray PX. Label this meeting point S.
6. The completed figure PQRS is the quadrilateral you needed to construct.

In simple words: First, calculate the missing angle S. Then, draw PQ as the base. Make the angles at P and Q. Mark R along the Q angle line. Then, make the angle at R, and where this line meets the P angle line, mark S.

🎯 Exam Tip: Always start by calculating any missing angle using the sum of angles property (\( 360^\circ \)) for a quadrilateral; this helps in planning your construction.