OP Malhotra Class 9 Maths Solutions Chapter 11 Rectilinear Figures Exercise 11 (C)

S Chand Class 9 ICSE Maths Solutions Chapter 11 Rectilinear Figures Ex 11(C)

Question 1. Construct a quadrilateral ABCD, when AB = 4 cm, BC = 6 cm, CD = DA = 5.2 cm and AC = 8 cm.
Answer:
Steps of construction:
(i) First, draw a line segment AB that is 4 cm long.
(ii) With A as the center and a radius of 8 cm, draw an arc. Then, with B as the center and a radius of 6 cm, draw another arc. These two arcs will meet at point C.
(iii) Connect points A to C and B to C with straight lines.
(iv) Next, with A as the center and a radius of 5.2 cm, draw an arc. With C as the center and a radius of 5.2 cm, draw a second arc. These two arcs will intersect at point D.
(v) Finally, join points A to D and C to D with straight lines. This completes the quadrilateral ABCD. Quadrilaterals are four-sided polygons, and their construction often relies on triangles formed by diagonals.
In simple words: Start by drawing the base AB. Then use a compass from A and B to find point C. After that, use A and C to find point D. Connect all the points to finish the shape.

🎯 Exam Tip: Always draw a rough sketch first to visualize the quadrilateral and plan the construction steps. Use a sharp pencil and precise compass measurements for accuracy.

Question 2. Construct a quadrilateral PQRS, where PQ = 3.5 cm, QR = 5.5 cm, QS = 5.5 cm, PS = 4.5 cm and SR = 4.5 cm. Measure PR.
Answer:
Steps of construction:
(i) Begin by drawing a line segment PQ that is 3.5 cm long.
(ii) With P as the center and a radius of 4.5 cm, draw an arc. Then, with Q as the center and a radius of 5.5 cm, draw another arc. These arcs will intersect at point S.
(iii) Join QS (already drawn from step ii) and RS. (Correction: The source step (iii) says "Join RS and QS". QS is formed by arcs from P and Q, so it just needs joining. RS comes from step (iv)).
(iv) Now, with S as the center and a radius of 4.5 cm, draw an arc. With Q as the center and a radius of 5.5 cm, draw another arc. These arcs will intersect at point R.
(v) Connect points P to R, Q to R, and R to S with straight lines. This forms the required quadrilateral PQRS. When constructing shapes, starting with a known triangle (like PQS here) makes it easier to build the rest.
Upon measuring, the length of PR should be 7 cm.
In simple words: Start with PQ. Use compasses from P and Q to find S. Then use S and Q to find R. Draw all the lines, and then measure the line PR.

🎯 Exam Tip: When constructing a quadrilateral with given sides and diagonals, always start by constructing one of the triangles formed by a diagonal, as this provides a solid base for the rest of the figure. Remember to measure the final requested length accurately.

Question 3. Construct a quadrilateral ABCD, when AB = 5 cm, BC = 4.2 cm, AD = 3.3 cm. AC = 5.5 cm and BD = 4.8 cm. Measure side CD.
Answer:
Steps of construction:
(i) First, draw a line segment AB that is 5 cm long.
(ii) With A as the center and a radius of 5.5 cm, draw an arc. Then, with B as the center and a radius of 4.2 cm, draw another arc. These two arcs will intersect at point C.
(iii) Join points A to C and B to C with straight lines.
(iv) Next, with A as the center and a radius of 3.3 cm, draw an arc. With B as the center and a radius of 4.8 cm, draw a second arc. These arcs will intersect at point D.
(v) Join points A to D, B to D, and C to D with straight lines. This completes the quadrilateral ABCD. Using two diagonals in construction often simplifies finding the vertex positions.
Upon measuring, the length of CD should be 2.5 cm.
In simple words: Draw AB first. Use A and B to find C. Then use A and B again to find D. Join all the points to create the shape. Finally, measure the length of CD.

🎯 Exam Tip: Pay close attention to which diagonal is used to locate which vertex. A mistake in using the correct diagonal or side length can lead to an incorrect quadrilateral. Double-check all measurements before joining points.

Question 4. Construct a quadrilateral ABCD, in which AB = BC = 4 cm, AD = CD = 6 cm and∠ABC= 120°.
Answer:
Steps of construction:
(i) Start by drawing a line segment AB that is 4 cm long.
(ii) At point B, draw a ray BX such that it forms an angle of 120° with AB. On this ray BX, mark point C such that BC is 4 cm long.
(iii) With A as the center and a radius of 6 cm, draw an arc. Then, with C as the center and a radius of 6 cm, draw another arc. These two arcs will intersect at point D.
(iv) Join points A to D and C to D with straight lines. This completes the quadrilateral ABCD. Knowing angles and adjacent sides is key in such constructions.
In simple words: Draw AB first. At B, draw a line at 120 degrees and mark C on it. From A and C, draw arcs of 6 cm to find point D. Connect A to D and C to D to finish the shape.

🎯 Exam Tip: When constructing quadrilaterals with angles, use a protractor carefully to draw the specified angle. Ensure the lengths are marked accurately on the rays or arcs to form the correct vertices.

Question 5. Construct a quadrilateral ABCD, in which AB = 4.6 cm, BC = 6.5 cm, CD = 6 cm, ∠B = 130° and ∠C = 75°.
Answer:
Steps of construction:
(i) Begin by drawing a line segment BC that is 6.5 cm long.
(ii) At point B, draw a ray BX making an angle of 130° with BC. At point C, draw a ray CY making an angle of 75° with BC (on the same side of BC as BX). On ray BX, cut off AB = 4.6 cm. On ray CY, cut off CD = 6 cm.
(iii) Join points A to D with a straight line. This creates the required quadrilateral ABCD. Using two angles and three sides to construct a quadrilateral is a common method.
In simple words: First, draw the line BC. At B, draw a line at 130 degrees and mark A on it. At C, draw another line at 75 degrees and mark D on it. Finally, connect A and D.

🎯 Exam Tip: When given angles, make sure to draw them accurately with a protractor. The direction of the rays for constructing angles (e.g., above or below the base line) is crucial for the final shape of the quadrilateral.

Question 6. Construct a quadrilateral PQRS, in which ∠Q= 45°, ∠R = 90°, QR = 5 cm, PQ = 9 cm and RS = 7 cm.
Answer:
Note: This figure cannot be drawn correctly with the given measurements. For example, if we draw QR and the angles at Q and R, the lengths of PQ and RS might not meet at suitable points.
Steps of construction:
(i) Draw a line segment QR that is 5 cm long.
(ii) At point Q, draw a ray QX making an angle of 45° with QR.
(iii) At point R, draw a ray RY making an angle of 90° with QR. Cut off RS = 7 cm on this ray RY.
(iv) Try to join point P (from QX, where PQ=9cm) to S. It will be challenging to meet these conditions simultaneously. This highlights that not all combinations of side lengths and angles will form a valid quadrilateral.
In simple words: Draw the base QR. Draw a 45-degree line from Q and a 90-degree line from R. Mark S on the R line. Then try to find P on the Q line such that PQ is 9 cm. This figure won't work out perfectly.

🎯 Exam Tip: Sometimes, a quadrilateral cannot be constructed with the given measurements because they are inconsistent. In such cases, state clearly that construction is not possible and explain why (e.g., sides not long enough, or angles not fitting the shape).

Question 7. Construct a quadrilateral ABCD, in which AB = 3.5 cm, BC = 6.5 cm.∠A = 75°, ∠B = 105° and ∠C = 120°.
Answer:
Steps of construction:
(i) Draw a line segment AB that is 3.5 cm long.
(ii) At point A, draw a ray AX making an angle of 75° with AB. At point B, draw a ray BY making an angle of 105° with AB.
(iii) On ray BY, cut off BC = 6.5 cm.
(iv) At point C, draw a ray CZ making an angle of 120° with BC. The point where ray CZ intersects ray AX will be point D. This forms the required quadrilateral ABCD. By finding three angles and two sides, the final vertex can be determined by the intersection of the last two rays.
In simple words: Draw AB. From A, draw a line at 75 degrees (AX). From B, draw a line at 105 degrees (BY) and mark C on it. From C, draw a line at 120 degrees (CZ). The point where CZ crosses AX will be D.

🎯 Exam Tip: When given multiple angles and sides, ensure the angles are drawn from the correct vertices and that the side lengths are marked accurately on the appropriate rays. The intersection of the final rays determines the last vertex.

Question 8. Construct a quadrilateral PQRS where PQ = 3.8 cm, QR = 6.8 cm, ∠P = ∠R = 105° and ∠S = 75°.
Answer:
Solution:
First, find the fourth angle: The sum of angles in a quadrilateral is 360°.
\( \angle Q = 360° - ( \angle P + \angle R + \angle S ) \)
\( \implies \angle Q = 360° - ( 105° + 105° + 75° ) \)
\( \implies \angle Q = 360° - 285° \)
\( \implies \angle Q = 75° \)
Steps of construction:
(i) Draw a line segment QR that is 6.8 cm long.
(ii) At point Q, draw a ray QX making an angle of 75° with QR. On this ray QX, cut off PQ = 3.8 cm.
(iii) At point R, draw a ray RY making an angle of 105° with QR. At point P, draw a ray PZ making an angle of 105° with PQ. The point where ray RY and ray PZ intersect will be S. This forms the required quadrilateral PQRS. Calculating the unknown angle first helps ensure all measurements are consistent before starting construction.
In simple words: First, find angle Q by subtracting the other angles from 360. Then, draw QR. At Q, draw a line at 75 degrees and mark P. At R, draw a line at 105 degrees. At P, draw another line at 105 degrees. Where the last two lines meet is point S.

🎯 Exam Tip: Always calculate any missing angles using the sum of angles in a quadrilateral (360°) before starting construction. This prevents issues if the provided angles are insufficient or if there's an internal inconsistency.

Question 9. Using ruler and compasses only construct the quadrilateral ABCD having given AB = 5 cm, BC = 2.5 cm, CD = 6 cm, ∠BAD = 90° and the diagonal AC = 5.5 cm.
Answer:
Steps of construction:
(i) Draw a line segment AB that is 5 cm long.
(ii) With A as the center and a radius of 5.5 cm, draw an arc. Then, with B as the center and a radius of 2.5 cm, draw another arc. These two arcs will intersect at point C.
(iii) Join points A to C and B to C with straight lines.
(iv) At point A, draw a ray AX making an angle of 90° with AB. (This forms ∠BAD).
(v) With C as the center and a radius of 6 cm, draw an arc that intersects ray AX at point D. This completes the quadrilateral ABCD. Using compasses and a ruler for 90-degree angles means constructing a perpendicular line, which adds precision.
In simple words: Draw AB. Use A and B to find C. Draw a 90-degree line from A (AX). From C, draw an arc of 6 cm that crosses AX to find D. Connect all points.

🎯 Exam Tip: When constructing a 90° angle using only ruler and compasses, remember the standard construction method of drawing arcs from a point on a line. This ensures accuracy without a protractor.

Question 10. Draw a quadrilateral ABCD with AB = 6 cm, BC = 4 cm, CD = 4 cm and∠ABC = ∠BCD = 90°.
Answer:
Steps of construction:
(i) Draw a line segment BC that is 4 cm long.
(ii) At point B, draw a ray BX making an angle of 90° with BC. At point C, draw a ray CY making an angle of 90° with BC.
(iii) From ray CY, cut off CD = 4 cm. From ray BX, cut off BA = 6 cm.
(iv) Join points A to D with a straight line. This completes the quadrilateral ABCD. This type of quadrilateral with two right angles and parallel sides is a trapezoid, or a rectangle/square if all sides were equal or parallel pairs.
In simple words: Draw BC. From B, draw a straight up line (90 degrees) and mark A on it. From C, draw another straight up line (90 degrees) and mark D on it. Connect A and D.

🎯 Exam Tip: When constructing a quadrilateral with two adjacent right angles, drawing the base first and then the perpendicular rays from its endpoints simplifies the process. Carefully mark the lengths along these rays before joining the final two points.