RS Aggarwal Class 8 Mathematics Solutions Chapter 17 Construction of Quadrilaterals

Exercise 17A

Question 1. Draw AB = 4.2 cm. With A as the centre and radius equal to 8 cm, draw an arc. With B as the centre and radius equal to 6 cm, draw another arc, cutting the previous arc at C. With A as the centre and radius equal to 5 cm, draw an arc. With C as the centre and radius equal to 5.2 cm, draw another arc, cutting the previous arc at D. Join AD and CD.
Answer: Follow the steps provided to construct quadrilateral ABCD. First, establish line segment AB measuring 4.2 cm. From point A, mark an arc using an 8 cm radius. Simultaneously, from point B, create an arc with a 6 cm radius until it crosses the first arc at point C. Next, from A, draw another arc with a 5 cm radius. From C, sketch an arc having a 5.2 cm radius so it intersects the previous arc at D. Finally, connect points A to D and C to D to form the quadrilateral. Thus, ABCD becomes the constructed quadrilateral.
In simple words: Use compass arcs from different centres with given measurements to find points C and D, then join them to complete the quadrilateral.

Exam Tip: Keep all measurements precise; even small errors in arc placement will shift the final shape significantly. Always verify that intersection points occur where arcs actually cross.

Question 2. Draw PQ = 5.4 cm. With P as the centre and radius equal to 4 cm, draw an arc. With Q as the centre and radius equal to 4.6 cm, draw another arc, cutting the previous arc at R. Join QR. With P as the centre and radius equal to 3.5 cm, draw an arc. With R as the centre and radius equal to 4.3 cm, draw another arc, cutting the previous arc at S. Join PS and RS.
Answer: Build the quadrilateral PQRS by first drawing segment PQ of length 5.4 cm. From P, create an arc with a 4 cm radius. From Q, draw an arc using a 4.6 cm radius to intersect the first arc at point R. Connect Q to R. Then from P, sketch another arc with a 3.5 cm radius. From R, draw an arc having a 4.3 cm radius until it crosses this new arc at S. Join P to S and R to S. Therefore, PQRS becomes the required quadrilateral.
In simple words: Create two intersection points, first R then S, by carefully positioning arcs from marked centres using the given measurements. Connect all points in sequence.

Exam Tip: Double-check that your arcs actually intersect; if they don't, your radius settings may be incorrect. Mark intersection points clearly before proceeding to the next step.

Question 3. Draw AB = 3.5 cm. With B as the centre and radius equal to 5.6 cm, draw an arc. With A as the centre and radius equal to 4.5 cm, draw another arc, cutting the previous arc at D. Join BD and AD. With D as the centre and radius equal to 4.5 cm, draw an arc. With B as the centre and radius equal to 3.8 cm, draw another arc, cutting the previous arc at C. Join BC and CD.
Answer: Construct quadrilateral ABCD beginning with line segment AB measuring 3.5 cm. Centred at B with a 5.6 cm radius, mark an arc. From A using a 4.5 cm radius, create another arc that intersects the first at point D. Connect B and D, and A and D. Now, with D as centre and a 4.5 cm radius, sketch a fresh arc. From B with a 3.8 cm radius, draw another arc crossing this one at C. Join the points B to C and C to D. Thus, ABCD is the required quadrilateral.
In simple words: Make point D first using two arcs, then make point C using two more arcs from different centres with different radii. Connect all points to complete the shape.

Exam Tip: This problem involves four arcs total - keep track of which centre and radius you use for each arc to avoid confusion.

Question 4. Draw AB = 3.6 cm. With B as the centre and radius equal to 4 cm, draw an arc. With A as the centre and radius equal to 2.7 cm, draw another arc, cutting the previous arc at D. Join BD and AD. With A as the centre and radius equal to 4.6 cm, draw an arc. With B as the centre and radius equal to 3.3 cm, draw another arc, cutting the previous arc at C. Join AC, BC and CD.
Answer: Start by drawing segment AB of length 3.6 cm. Using B as the centre with a 4 cm radius, mark an arc. From A using a 2.7 cm radius, create another arc crossing the first at D. Connect B to D and A to D. Next, from A with a 4.6 cm radius, sketch a new arc. From B using a 3.3 cm radius, draw another arc to meet this one at C. Join A to C, B to C, and C to D. Thus, ABCD is the required quadrilateral.
In simple words: Use two pairs of arcs from two different base points to locate the two remaining vertices D and C. Link all points appropriately.

Exam Tip: Always join the vertices in the correct order after locating them; label each point clearly on your diagram.

Question 5. Draw QR = 7.5 cm. With Q as the centre and radius equal to 10 cm, draw an arc. With R as the centre and radius equal to 5 cm, draw another arc, cutting the previous arc at S. Join QS and RS. With S as the centre and radius equal to 6 cm, draw an arc. With R as the centre and radius equal to 6 cm, draw another arc, cutting the previous arc at P. Join PS and PR. Also, FQ = 4.9 cm.
Answer: Begin with segment QR having length 7.5 cm. Centred at Q with a 10 cm radius, mark an arc. From R using a 5 cm radius, sketch another arc to intersect the first at S. Connect Q to S and R to S. Now from S with a 6 cm radius, draw a new arc. Using R as centre with a 6 cm radius, create another arc meeting this one at P. Join S to P and R to P. The measurement FQ equals 4.9 cm. Thus, PQRS is the required quadrilateral.
In simple words: Locate point S first using arcs from Q and R, then locate P using arcs from S and R. Connect them properly to form the complete shape.

Exam Tip: When multiple arcs are involved, label each temporary point right away so you don't lose track of which intersection is which.

Question 6. Draw AB = 3.4 cm. With B as the centre and radius equal to 4 cm, draw an arc. With A as the centre and radius equal to 5.7 cm, draw another arc, cutting the previous arc at D. Join BD and AD. With A as the centre and radius equal to 8 cm, draw an arc. With D as the centre and radius equal to 3 cm, draw another arc, cutting the previous arc at C. Join AC, CD and BC.
Answer: Draw line segment AB measuring 3.4 cm. From B with a 4 cm radius, mark an arc. From A using a 5.7 cm radius, create another arc intersecting the first at D. Join B to D and A to D. Then from A with an 8 cm radius, draw a fresh arc. Using D as centre with a 3 cm radius, sketch another arc crossing this one at C. Connect A to C, C to D, and B to C. Thus, ABCD is the required quadrilateral.
In simple words: First find point D, then find point C using different arc pairs. Make sure to connect all vertices in the right order.

Exam Tip: Before connecting points, verify that you have located all four vertices correctly by reviewing your arc measurements.

Question 7. Draw AB = 3.6 cm. With A as the centre and radius equal to 4 cm, draw an arc. With B as the centre and radius equal to 5 cm, draw another arc, cutting the previous arc at D. Join AD and BD. With B as the centre and radius equal to 4.6 cm, draw an arc. With D as the centre and radius equal to 5.2 cm, draw another arc, cutting the previous arc at C. Join BC and DC.
Answer: Start by drawing AB of length 3.6 cm. Using A as the centre with a 4 cm radius, mark an arc. From B with a 5 cm radius, create another arc to cross the first at D. Connect A to D and B to D. Next, from B with a 4.6 cm radius, sketch a new arc. Using D as centre with a 5.2 cm radius, draw another arc meeting this one at C. Join B to C and D to C. Thus, ABCD is the required quadrilateral.
In simple words: Find D first by intersecting two arcs, then find C by intersecting two more arcs from different centres. Connect all four points.

Exam Tip: Keep your compass settings steady for each arc; any slip will throw off your intersection points significantly.

Question 8. Draw AB = 3.5 cm. Make angle ABC = 120°. With B as the centre, draw an arc 3.5 cm and name that point C. With C as the centre, draw an arc 5.2 cm. With A as the centre, draw another arc 5.2 cm, cutting the previous arc at D. Join CD and AD.
Answer: Draw segment AB of length 3.5 cm. At B, construct an angle of 120°. Using B as the centre with a 3.5 cm radius, mark point C on this angle line. From C using a 5.2 cm radius, sketch an arc. From A with a 5.2 cm radius, create another arc crossing the previous one at D. Join C to D and A to D. Thus, ABCD is the required quadrilateral.
In simple words: Start with AB and make a 120° angle at B to place C. Then use equal radii from both A and C to find D.

Exam Tip: Angle construction must be precise; use a protractor if your compass cannot measure angles directly.

Question 9. Draw AB = 2.9 cm. Make angle A = 70°. With A as the centre, draw an arc of 3.4 cm. Name that point as D. With D as the centre, draw an arc of 2.7 cm. With B as the centre, draw an arc of 3.2 cm, cutting the previous arc at C. Join CD and BC.
Answer: Begin by sketching line segment AB measuring 2.9 cm. At A, construct an angle of 70°. Using A as the centre with a 3.4 cm radius, place point D. From D with a 2.7 cm radius, mark an arc. From B using a 3.2 cm radius, create another arc to intersect the previous one at C. Join D to C and B to C. Then, ABCD is the required quadrilateral.
In simple words: Make the angle at A first, then place D on that angle line. Use two more arcs from D and B to find C.

Exam Tip: When measuring the angle, be as accurate as possible; small angle errors lead to misaligned sides in the final shape.

Question 10. Draw BC = 5 cm. Make angle B = 125° and angle C = 60°. With B as the centre, draw an arc of 3.5 cm. Name that point as A. With C as the centre, draw an arc of 4.6 cm. Name that point as D. Join A and D.
Answer: Construct line segment BC of length 5 cm. At B, form an angle of 125°. At C, form an angle of 60°. Using B as the centre with a 3.5 cm radius, locate point A on the angle line from B. Using C as the centre with a 4.6 cm radius, locate point D on the angle line from C. Join A to D. Thus, ABCD is the required quadrilateral.
In simple words: Build two angles at the ends of BC, then place A and D using compass arcs on these angle lines. Connect them to finish.

Exam Tip: Two angles at known vertices will guide the placement of the other two vertices; mark these angles clearly.

Question 11. Draw QR = 5.6 cm. Make angle Q = 45° and angle R = 90°. With Q as the centre, draw an arc of 6 cm. Name that point as P. With R as the centre, draw an arc of 2.7 cm. Name that point as S. Join P and S.
Answer: Start by drawing line segment QR with length 5.6 cm. At Q, construct an angle of 45°. At R, construct an angle of 90°. From Q using a 6 cm radius, mark point P on the angle line at Q. From R using a 2.7 cm radius, mark point S on the angle line at R. Join P to S. Therefore, PQRS is the required quadrilateral. Note that the angle calculations show \( \angle A + \angle B + \angle C + \angle D = 360° \); \( 50° + 105° + \angle C + 80° = 360° \); \( 235° + \angle C = 360° \); \( \angle C = 125° \).
In simple words: Form the two specified angles at Q and R, place P and S using compass measurements on the angle lines, then connect them.

Exam Tip: The sum of all interior angles in any quadrilateral always equals 360° - use this to check your angle calculations.

Question 12. Draw PQ = 5 cm. From the angle sum property, \( \angle P + \angle Q + \angle R + \angle S = 360° \); \( 100° + \angle Q + 100° + 75° = 360° \); \( 275° + \angle Q = 360° \); \( \angle Q = 85° \). Make angle P = 100° and angle Q = 85°. With Q as the centre, draw an arc of 6.5 cm. Make angle R = 100°. Join R and S. Measure angle S = 75°.
Answer: Draw line segment PQ measuring 5 cm. Use the angle sum property of quadrilaterals: \( 100° + \angle Q + 100° + 75° = 360° \), which gives \( \angle Q = 85° \). At P, form an angle of 100°. At Q, form an angle of 85°. From Q with a 6.5 cm radius, mark an arc to place R. At R, construct an angle of 100°. Connect R and S to finish the quadrilateral. When measured, \( \angle S = 75° \). Thus, PQRS is the required quadrilateral.
In simple words: Calculate the missing angle using the 360° rule, then construct the three known angles at their respective vertices and use arc measurements to place the points.

Exam Tip: Always verify that your four angles sum to exactly 360° - any deviation signals an error in construction or measurement.

Question 13. Draw AB = 4 cm. Make angle B = 90°. Using the Pythagorean theorem: \( AC^2 = AB^2 + BC^2 \); \( 5^2 = 4^2 + BC^2 \); \( 25 - 16 = BC^2 \); \( BC = 3 \) cm. Make angle C = 90°. With A as the centre and radius equal to 5.5 cm, draw an arc and name that point as D.
Answer: Draw segment AB of length 4 cm. At B, construct a right angle (90°). Apply the Pythagorean theorem to find BC: since \( AC^2 = AB^2 + BC^2 \), we have \( 5^2 = 4^2 + BC^2 \), giving \( BC = 3 \) cm. At C, also construct a right angle (90°). From A using a radius of 5.5 cm, mark an arc to place point D. Thus, ABCD is the required quadrilateral.
In simple words: Calculate BC using the Pythagorean formula with the known side AC. Build right angles at both B and C, then use a compass arc from A to find the last vertex.

Exam Tip: When a right angle is mentioned, use it to your advantage - it often triggers the Pythagorean theorem for finding missing sides.

Exercise 17B

Question 1. Draw AB = 5.2 cm. With B as the centre, draw an arc of 4.7 cm. With A as the centre, draw another arc of 7.6 cm, cutting the previous arc at C. Join A and C. We know that the opposite sides of a parallelogram are equal. Thus, with C as the centre, draw an arc of 5.2 cm. With A as the centre, draw another arc of 4.7 cm, cutting the previous arc at D. Join CD and AD.
Answer: Begin by drawing segment AB measuring 5.2 cm. From B using a 4.7 cm radius, create an arc. From A with a 7.6 cm radius, sketch another arc to intersect the first at C. Connect A to C. Since opposite sides of a parallelogram are equal, from C using a 5.2 cm radius, mark an arc. From A with a 4.7 cm radius, draw another arc crossing the first at D. Join C to D and A to D. Thus, ABCD becomes the required parallelogram.
In simple words: Find C using arcs from A and B, then use the equal-sides property to find D using arcs from C and A. All angles will automatically fall into place.

Exam Tip: The key property of a parallelogram is that opposite sides are equal - use this to avoid calculating angles separately.

Question 2. Draw AB = 4.3 cm. With B as the centre, draw an arc of 6.8 cm. With A as the centre, draw another arc of 4 cm, cutting the previous arc at D. Join BD and AD. We know that the opposite sides of a parallelogram are equal. Thus, with D as the centre, draw an arc of 4.3 cm. With B as the centre, draw another arc of 4 cm, cutting the previous arc at C. Join CD and BC.
Answer: Start by drawing line segment AB of length 4.3 cm. From B with a 6.8 cm radius, sketch an arc. From A using a 4 cm radius, mark another arc that crosses the first at D. Connect B to D and A to D. By the property that opposite sides of a parallelogram are equal, from D with a 4.3 cm radius, create an arc. From B with a 4 cm radius, draw another arc intersecting the first at C. Join D to C and B to C. Thus, ABCD is the required parallelogram.
In simple words: First locate D, then use the equal-opposite-sides rule to find C by mirroring the arc measurements from the other end.

Exam Tip: When constructing a parallelogram, always apply the equal opposite-sides property after placing the first two adjacent vertices.

Question 3. Draw PQ = 4 cm. Make angle PQR = 60°. With Q as the centre, draw an arc of 6 cm and name that point as R. With R as the centre, draw an arc of 4 cm and name that point as S. Join SR and PS.
Answer: Draw line segment PQ of length 4 cm. At Q, construct an angle of 60°. Using Q as the centre with a 6 cm radius, place point R on the angle line. From R with a 4 cm radius, mark point S. Connect S to R and P to S. Thus, PQRS is the required parallelogram.
In simple words: Build the 60° angle at Q, place R using a compass measurement on the angle line, then place S using the equal-side property.

Exam Tip: When an angle and two side lengths are given, the angle must be at one of the known vertices for the construction to proceed smoothly.

Question 5. We know that the diagonals of a parallelogram bisect each other. Draw AB = 4.4 cm. With A as the centre and radius 2.8 cm, draw an arc. With B as the centre and radius 3.5 cm, draw another arc, cutting the previous arc at point O. Join OA and OB. Produce OA to C, such that OC = AO. Produce OB to D, such that OB = OD. Join AD, BC, and CD.
Answer: Begin with line segment AB measuring 4.4 cm. From A using a 2.8 cm radius, sketch an arc. From B with a 3.5 cm radius, create another arc intersecting the first at O. Connect O to both A and B. Using the property that diagonals bisect each other, extend line OA beyond A to C such that OC equals AO. Extend line OB beyond B to D such that OD equals OB. Join A to D, B to C, and C to D. Thus, ABCD is the required parallelogram, and the other side measures 4.5 cm in length.
In simple words: Find the midpoint O of the diagonals first, then create the vertices by extending equal distances on both sides of O.

Exam Tip: The diagonal bisection property is powerful - if you find the midpoint, the other two vertices follow automatically by equal extension.

Question 6. Draw AB = 6.5 cm. Draw a perpendicular at point A. Name that ray as AX. From point A, draw an arc of length 2.5 cm on the ray AX and name that point as L. On point L, make a perpendicular. Draw a straight line YZ passing through L, which is perpendicular to the ray AX. Cut an arc of length 3.4 cm on the line YZ and name it as C. From point C, cut an arc of length 6.5 cm on the line YZ. Name that point as D. Join BC and AD.
Answer: Draw segment AB of length 6.5 cm. At A, create a perpendicular ray AX. From A along ray AX, mark a point L at a distance of 2.5 cm. At L, erect another perpendicular. Construct a straight line YZ through L that is perpendicular to ray AX. On line YZ, cut an arc of 3.4 cm from L and name that point as C. From C, mark an arc of 6.5 cm on line YZ to locate D. Join B to C and A to D. Therefore, quadrilateral ABCD is a parallelogram. The altitude from C measures 2.5 cm in length.
In simple words: Build perpendiculars at A and L to create right angles. Use these perpendicular lines to place C and D at fixed distances, then connect all vertices.

Exam Tip: When perpendiculars are involved, use a set square for accuracy - it ensures your right angles are truly 90 degrees.

Question 8. We know that the diagonals of a parallelogram bisect each other. Draw AC = 3.8 cm. Bisect AC at O. Make angle COX = 60°. Produce XO to Y. From O: \( OB = \frac{1}{2} \left( 4.6 \right) \) cm = 2.3 cm and \( OD = \frac{1}{2} \left( 4.6 \right) \) cm = 2.3 cm. Join AB, BC, CD and AD.
Answer: Draw line segment AC of length 3.8 cm. Find the midpoint O of AC by bisecting it. At O, construct an angle of 60° with the line segment, and extend it to form line XY. From O, calculate that \( OB = \frac{1}{2}(4.6) = 2.3 \) cm and \( OD = \frac{1}{2}(4.6) = 2.3 \) cm. Mark points B and D on line XY at these distances from O on opposite sides. Connect A to B, B to C, C to D, and D to A. Thus, ABCD is the required parallelogram, where AC and BD are the diagonals of length 3.8 cm and 4.6 cm respectively.
In simple words: Find the centre of one diagonal, then mark the other two vertices at equal distances on a perpendicular line. The diagonal-bisection property guarantees a parallelogram.

Exam Tip: The diagonals of a parallelogram always bisect each other at the same point - use this to construct vertices quickly from known diagonals.

Question 10. All the sides of a square are equal. Draw AB = 6.4 cm. Make angle A = 90°. Draw an arc of length 6.4 cm from point A and name that point as D. Draw an arc of length 6.4 cm from point B and name that point as C. Join C and D.
Answer: Begin by drawing line segment AB of length 6.4 cm. At A, construct a right angle (90°). From A using a 6.4 cm radius, place point D. From B with a 6.4 cm radius, locate point C. Join C and D. Thus, ABCD is the required square, since all sides measure 6.4 cm and all angles are right angles.
In simple words: In a square, all sides are equal and all angles are 90°. Mark arcs of equal length from both A and B, then connect the two new points.

Exam Tip: A square has the simplest construction among all quadrilaterals - just make one right angle and two equal arcs of the same length.

Question 11. We know that the diagonals of a square bisect each other at right angles. Draw AC = 5.8 cm. Draw the perpendicular bisector XY of AC, meeting it at O. From O: \( OB = \frac{1}{2} \left( 5.8 \right) \) cm = 2.9 cm and \( OD = \frac{1}{2} \left( 5.8 \right) \) cm = 2.9 cm. Join AB, BC, CD and DA. ABCD is the required square. The other side is 4.8 cm in length.
Answer: Draw diagonal AC of length 5.8 cm. Construct the perpendicular bisector XY of AC, intersecting it at midpoint O. From O, calculate \( OB = \frac{1}{2}(5.8) = 2.9 \) cm and \( OD = \frac{1}{2}(5.8) = 2.9 \) cm. Mark points B and D on the perpendicular bisector at these distances from O, on opposite sides. Join all adjacent vertices: A to B, B to C, C to D, and D to A. Thus, ABCD is the required square. Note that the other side measures 4.8 cm in length.
In simple words: Find the midpoint and perpendicular to one diagonal. Place the other two vertices equidistant from this midpoint on the perpendicular line. This automatically forms a square.

Exam Tip: When both diagonals are given or one is known, the perpendicular bisector method is the fastest way to construct a square.

Question 13. Draw AB = 4 cm. With B as the centre, draw an arc of 4 cm. With A as the centre, draw another arc of 6.5 cm, cutting the previous arc at C. Join AC and BC. With C as the centre, draw an arc of 4 cm. With A as the centre, draw another arc of 4 cm, cutting the previous arc at D. Join AD and CD.
Answer: Start by drawing segment AB of length 4 cm. From B using a 4 cm radius, create an arc. From A with a 6.5 cm radius, sketch another arc crossing the first at C. Connect A to C and B to C. Next, from C with a 4 cm radius, mark an arc. From A with a 4 cm radius, draw another arc intersecting the first at D. Join A to D and C to D. Thus, ABCD is the required rhombus, as shown in the figure.
In simple words: Place C using two arcs from A and B, then place D using two arcs from A and C. The equal sides create the rhombus shape.

Exam Tip: In a rhombus, all four sides are equal - check that your arc radii reflect this property.

Question 15. Draw AB = 7.2 cm. Draw angle ABY = 60°. Draw angle BAX = 120°. Sum of the adjacent angles is 180°. \( \angle BAX + \angle ABY = 180° \) → \( 120° + \angle ABY = 180° \) → \( \angle BAX = 180° - 60° = 120° \). Set off AD (7.2 cm) along AX and BC (7.2 cm) along BY. Join C and D.
Answer: Draw line segment AB measuring 7.2 cm. At B, construct angle ABY = 60°. At A, construct angle BAX = 120°. Note that these adjacent angles sum to 180°, which is the property of a trapezoid with two given angles. Along ray AX, mark point D at a distance of 7.2 cm from A. Along ray BY, mark point C at a distance of 7.2 cm from B. Join C to D. Thus, ABCD is the required rhombus as indicated in the figure. The angles shown are 120° and 60°, which are supplementary.
In simple words: Build two angles at the endpoints of AB, then extend equal-length sides along these angle lines. Connect the endpoints to complete the shape.

Exam Tip: When adjacent angles in a quadrilateral sum to 180°, the opposite sides are parallel - this indicates a trapezoid or parallelogram.

Question 16. Draw AB = 6 cm. Make angle ABX = 75°. With B as the centre, draw an arc at 4 cm. Name that point as C. Since AB || CD, ∴ \( \angle ABX + \angle BCY = 180° \) → \( \angle BCY = 180° - 75° = 105° \). Make angle BCY, equal to 105°. At C, draw an arc of length 3.2 cm, cutting the previous arc at D. Join A and D.
Answer: Draw line segment AB of length 6 cm. At B, construct angle ABX = 75°. Using B as the centre with a 4 cm radius, place point C. Since AB is parallel to CD, the co-interior angles are supplementary: \( \angle ABX + \angle BCY = 180° \), so \( \angle BCY = 105° \). At C, construct this angle equal to 105°. From C, mark an arc of length 3.2 cm intersecting the angle line at D. Join A to D. Thus, ABCD is the required trapezium, as shown in the diagram.
In simple words: Build angles at B and C such that they sum to 180°, then mark C and D using compass arcs from B and C. This ensures the two sides are parallel.

Exam Tip: For a trapezium with parallel sides, use the co-interior angle property - angles on the same side of a transversal sum to 180°.