CONSTRUCTION OF POLYGONS - Questions & Answers
EXERCISE 141. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) A quadrilateral can be constructed, if its :
(i) three sides and one angle are known
(ii) four sides and one angle are known
(iii) two sides and two angles are known
(iv) four sides are known
Answer: (ii) four sides and one angle are known
(b) A parallelogram can be constructed, if its:
(i) all the four sides are known
(ii) opposite sides are known
(iii) opposite angles are known
(iv) two adjacent sides and contained angle are known
Answer: (iv) two adjacent sides and contained angle are known
(c) A parallelogram can be constructed, if its:
(i) both the diagonals and included angle are given
(ii) both the diagonals and one angle are given
(iii) both the diagonals are given
(iv) one side and one diagonal are given
Answer: (i) both the diagonals and included angle are given
(d) A trapezium can be constructed, if its :
(i) parallel sides are given
(ii) non-parallel sides are given
(iii) both the non-parallel sides and both the parallel sides are given
(iv) four sides are given
Answer: (iv) four sides are given
(e) A rhombus can be constructed, if its :
(i) one side is given
(ii) one side and one diagonal are given
(iii) one of the diagonals is given
(iv) opposite angles are given
Answer: (ii) one side and one diagonal are given
Construct a quadrilateral ABCD, when :
2. AB = 3.2 cm, BC = 5.2 cm, CD = 6.2 cm, DA = 4.2 cm and BD = 5.2 cm.
Answer Steps:
1. Draw line segment BD = 5.2 cm.
2. With B as centre and radius 3.2 cm, draw an arc.
3. With D as centre and radius 4.2 cm, draw another arc cutting the previous arc at A.
4. Join AB and AD to form triangle ABD.
5. With B as centre and radius 5.2 cm, draw an arc on the other side of BD.
6. With D as centre and radius 6.2 cm, draw another arc cutting the previous arc at C.
7. Join BC and DC.
8. ABCD is the required quadrilateral.
3. AB = 7.2 cm, BC = 5.8 cm, CD = 6.3 cm, AD = 4.3 cm and angle A = 75°.
Answer Steps:
1. Draw line segment AB = 7.2 cm.
2. At point A, construct an angle of 75° using a compass or protractor.
3. From this angle ray, cut off AD = 4.3 cm.
4. With D as centre and radius 6.3 cm, draw an arc.
5. With B as centre and radius 5.8 cm, draw another arc intersecting the previous arc at C.
6. Join DC and BC.
7. ABCD is the required quadrilateral.
4. Angle A = 90°, AB = 4.6 cm, BD = 6.4 cm, AC = 6.0 cm and CD = 4.2 cm.
Answer Steps:
1. Draw AB = 4.6 cm.
2. At point A, construct an angle of 90°.
3. With B as centre and radius 6.4 cm, draw an arc cutting the perpendicular line from A at D.
4. Join BD and AD.
5. With A as centre and radius 6.0 cm, draw an arc.
6. With D as centre and radius 4.2 cm, draw another arc intersecting the previous arc at C.
7. Join BC and DC.
8. ABCD is the required quadrilateral.
5. AB = 3.8 cm, AC = 4.8 cm, AD = 2.8 cm, angle A = 105° and angle B = 60°.
Answer Steps:
1. Draw AB = 3.8 cm.
2. At point A, construct an angle of 105°.
3. From this line, cut off AD = 2.8 cm.
4. At point B, construct an angle of 60°.
5. With A as centre and radius 4.8 cm, draw an arc cutting the 60° angle line from B at point C.
6. Join DC and AC.
7. ABCD is the required quadrilateral.
6. BC = 7.5 cm, AC = 5.8 cm, AD = 3.6 cm, CD = 4.2 cm and angle A = 120°.
Answer Steps:
1. First, construct triangle ADC: Draw AD = 3.6 cm.
2. With A as centre and radius 5.8 cm, draw an arc.
3. With D as centre and radius 4.2 cm, draw an arc cutting the previous arc at C.
4. Join AC and DC.
5. At point A, construct angle DAB = 120° relative to the line AD.
6. With C as centre and radius 7.5 cm, draw an arc cutting the angle ray from A at point B.
7. Join BC and AB.
8. ABCD is the required quadrilateral.
7. AD = AB = 4 cm, BC = 2.8 cm, CD = 2.5 cm and angle BAD = 45°.
Answer Steps:
1. Draw AB = 4 cm.
2. At point A, construct angle BAD = 45°.
3. Cut off AD = 4 cm from this ray.
4. With B as centre and radius 2.8 cm, draw an arc.
5. With D as centre and radius 2.5 cm, draw an arc intersecting the previous arc at C.
6. Join BC and DC.
7. ABCD is the required quadrilateral.
8. AB = 6.3 cm, BC = CD = 4.2 cm and ∠ABC = ∠BCD = 90°.
Answer Steps:
1. Draw BC = 4.2 cm.
2. At B, construct an angle of 90°.
3. Cut off AB = 6.3 cm on this perpendicular ray.
4. At C, construct an angle of 90°.
5. Cut off CD = 4.2 cm on this perpendicular ray.
6. Join AD.
7. ABCD is the required quadrilateral.
Construct a parallelogram ABCD, when :
9. AB = 4.4 cm, AD = 6.2 cm and AC = 4.8 cm.
Answer Steps:
1. Draw AB = 4.4 cm.
2. With A as centre and radius 4.8 cm, draw an arc.
3. With B as centre and radius 6.2 cm (since opposite sides of a parallelogram are equal, BC = AD = 6.2 cm), draw an arc cutting the previous arc at C.
4. Join AC and BC.
5. With A as centre and radius 6.2 cm, draw an arc.
6. With C as centre and radius 4.4 cm (CD = AB = 4.4 cm), draw an arc intersecting the previous arc at D.
7. Join AD and CD.
8. ABCD is the required parallelogram.
10. Diagonal AC = 6.4 cm, diagonal BD = 8.2 cm and angle between the diagonals = 60°.
Answer Steps:
1. Draw diagonal BD = 8.2 cm.
2. Draw the perpendicular bisector of BD to find its mid-point O. BO = OD = 4.1 cm.
3. Draw a line passing through O making an angle of 60° with BD.
4. From O, cut off OA = 3.2 cm and OC = 3.2 cm (half of AC = 6.4 cm) on opposite sides of O on this line.
5. Join AB, BC, CD, and DA.
6. ABCD is the required parallelogram.
11. AB = 5.8 cm, diagonal AC = 8.2 cm and diagonal BD = 6.2 cm.
Answer Steps:
1. Draw AB = 5.8 cm.
2. With A as centre and radius 4.1 cm (half of AC), draw an arc.
3. With B as centre and radius 3.1 cm (half of BD), draw an arc cutting the previous arc at O (the intersection point of diagonals).
4. Join AO and produce it to C such that OC = AO = 4.1 cm.
5. Join BO and produce it to D such that OD = BO = 3.1 cm.
6. Join BC, CD, and DA.
7. ABCD is the required parallelogram.
12. AB = 6.0 cm, AD = 5.0 cm and ∠A = 45°.
Answer Steps:
1. Draw AB = 6.0 cm.
2. At A, construct an angle of 45°.
3. Cut off AD = 5.0 cm on this ray.
4. With D as centre and radius 6.0 cm (DC = AB), draw an arc.
5. With B as centre and radius 5.0 cm (BC = AD), draw an arc intersecting the previous arc at C.
6. Join DC and BC.
7. ABCD is the required parallelogram.
13. Base AB = 6.5 cm, BC = 4 cm and the altitude corresponding to AB = 3.1 cm.
Answer Steps:
1. Draw base AB = 6.5 cm.
2. Construct a perpendicular at any point on AB and mark a point at a height of 3.1 cm.
3. Draw a line parallel to AB passing through this marked point.
4. With B as centre and radius 4 cm, draw an arc cutting the parallel line at C.
5. With A as centre and radius 4 cm, draw an arc cutting the parallel line at D on the same side.
6. Join AD and BC.
7. ABCD is the required parallelogram.
14. AB = 4.5 cm, ∠B = 120° and the distance between AB and DC = 3.0 cm.
Answer Steps:
1. Draw base AB = 4.5 cm.
2. Draw a line parallel to AB at a perpendicular distance of 3.0 cm.
3. At B, construct an angle of 120° and let this ray intersect the parallel line at C.
4. With C as centre and radius 4.5 cm (since CD = AB), draw an arc cutting the parallel line at D.
5. Join AD.
6. ABCD is the required parallelogram.
15. Base BC = 5.6 cm, diagonal BD = 6.5 cm and altitude = 3.2 cm.
Answer Steps:
1. Draw base BC = 5.6 cm.
2. Draw a line parallel to BC at a perpendicular distance of 3.2 cm.
3. With B as centre and radius 6.5 cm, draw an arc cutting the parallel line at D.
4. With D as centre and radius 5.6 cm (since AD = BC), cut an arc on the parallel line to find A.
5. Join AB, CD, and BD.
6. ABCD is the required parallelogram.
Construct a rectangle ABCD, when :
16. Its sides are 6.0 cm and 7.2 cm.
Answer Steps:
1. Draw base AB = 7.2 cm.
2. At points A and B, construct angles of 90°.
3. From these perpendicular rays, cut off AD = 6.0 cm and BC = 6.0 cm.
4. Join CD.
5. ABCD is the required rectangle.
17. One side = 4 cm and one diagonal is 5 cm. Measure the length of other side.
Answer Steps:
1. Draw base AB = 4 cm.
2. At B, construct an angle of 90°.
3. With A as centre and radius 5 cm, draw an arc cutting the perpendicular ray at C.
4. Measure the length of BC. (It will measure 3 cm).
5. With C as centre and radius 4 cm, draw an arc.
6. With A as centre and radius 3 cm, draw an arc intersecting the previous arc at D.
7. Join AD and CD.
8. ABCD is the required rectangle. The length of the other side is 3 cm.
18. One diagonal = 6.0 cm and the acute angle between the diagonals = 45°.
Answer Steps:
1. Draw diagonal AC = 6.0 cm.
2. Find its mid-point O by constructing a perpendicular bisector (AO = OC = 3.0 cm).
3. At O, draw a line making an angle of 45° with AC and extend it on both sides.
4. Since diagonals of a rectangle are equal and bisect each other, cut off OB = 3.0 cm and OD = 3.0 cm on this line.
5. Join AB, BC, CD, and DA.
6. ABCD is the required rectangle.
19. Area = 24 cm² and base = 4.8 cm.
Answer Steps:
1. Calculate height: Height = Area / base = 24 / 4.8 = 5 cm.
2. Draw base AB = 4.8 cm.
3. At A and B, construct angles of 90°.
4. Cut off AD = 5 cm and BC = 5 cm on these perpendiculars.
5. Join CD.
6. ABCD is the required rectangle.
20. Area = 36 cm² and height = 4.5 cm.
Answer Steps:
1. Calculate base: Base = Area / height = 36 / 4.5 = 8 cm.
2. Draw base AB = 8 cm.
3. At A and B, construct angles of 90°.
4. Cut off AD = 4.5 cm and BC = 4.5 cm on these perpendiculars.
5. Join CD.
6. ABCD is the required rectangle.
Construct a trapezium ABCD, when :
21. AB = 4.8 cm, BC = 6.8 cm, CD = 5.4 cm, angle B = 60° and AD // BC.
Answer Steps:
1. Draw BC = 6.8 cm.
2. At B, construct an angle of 60°.
3. Cut off AB = 4.8 cm on this ray.
4. Through A, draw a line parallel to BC.
5. With C as centre and radius 5.4 cm, draw an arc cutting the parallel line at D.
6. Join CD.
7. ABCD is the required trapezium.
22. AB = CD = 3.2 cm, BC = 6.0 cm, AD = 4.4 cm and AD // BC.
Answer Steps:
1. Draw BC = 6.0 cm.
2. Mark a point E on BC such that EC = AD = 4.4 cm. (This leaves BE = 6.0 - 4.4 = 1.6 cm).
3. Construct triangle ABE with AB = 3.2 cm, BE = 1.6 cm, and AE = CD = 3.2 cm.
4. From A, draw a line parallel to BC.
5. With A as centre and radius 4.4 cm, draw an arc cutting the parallel line at D.
6. Join CD.
7. ABCD is the required trapezium.
Construct a rhombus ABCD, when :
23. Its one side = 6 cm and ∠A = 60°.
Answer Steps:
1. Draw AB = 6 cm.
2. At A, construct an angle of 60°.
3. Cut off AD = 6 cm from this ray.
4. With D as centre and radius 6 cm, draw an arc.
5. With B as centre and radius 6 cm, draw an arc intersecting the previous arc at C.
6. Join DC and BC.
7. ABCD is the required rhombus.
24. One side = 5.4 cm and one diagonal is 7.0 cm.
Answer Steps:
1. Draw diagonal AC = 7.0 cm.
2. With A as centre and radius 5.4 cm, draw arcs on both sides of AC.
3. With C as centre and radius 5.4 cm, draw arcs intersecting the previous arcs at B and D.
4. Join AB, BC, CD, and DA.
5. ABCD is the required rhombus.
25. Diagonal AC = 6.3 cm and diagonal BD = 5.8 cm.
Answer Steps:
1. Draw diagonal AC = 6.3 cm.
2. Draw the perpendicular bisector of AC to find its mid-point O.
3. From O, cut off OB = 2.9 cm and OD = 2.9 cm (half of BD = 5.8 / 2 = 2.9 cm) on the perpendicular bisector line.
4. Join AB, BC, CD, and DA.
5. ABCD is the required rhombus.
26. One side = 5.0 cm and height = 2.6 cm.
Answer Steps:
1. Draw base AB = 5.0 cm.
2. Draw a line parallel to AB at a perpendicular distance of 2.6 cm (height).
3. With A as centre and radius 5.0 cm, draw an arc cutting the parallel line at D.
4. With B as centre and radius 5.0 cm, draw an arc cutting the parallel line at C.
5. Join AD and BC.
6. ABCD is the required rhombus.
27. ∠A = 60° and height = 3.0 cm.
Answer Steps:
1. Draw a base line AX.
2. At A, construct an angle of 60°.
3. Draw a line parallel to AX at a perpendicular height of 3.0 cm, intersecting the 60° angle ray at D.
4. Measure length AD. (In a rhombus, all sides are equal, so AB = AD).
5. With A as centre and radius equal to AD, mark B on AX.
6. With D as centre and radius equal to AD, mark C on the parallel line.
7. Join BC.
8. ABCD is the required rhombus.
28. Diagonal AC = 6.0 cm and height = 3.5 cm.
Answer Steps:
1. Draw a pair of parallel lines separated by a perpendicular distance of 3.5 cm.
2. Take a point A on the lower parallel line.
3. With A as centre and radius 6.0 cm, draw an arc cutting the upper parallel line at C.
4. Draw the perpendicular bisector of AC. Let it meet the lower line at B and the upper line at D.
5. Join AB, BC, CD, and DA.
6. ABCD is the required rhombus.
Construct a square ABCD, when :
29. One side = 4.5 cm.
Answer Steps:
1. Draw AB = 4.5 cm.
2. At A and B, construct angles of 90°.
3. Cut off AD = 4.5 cm and BC = 4.5 cm from the perpendicular rays.
4. Join CD.
5. ABCD is the required square.
30. One diagonal = 5.4 cm.
Answer Steps:
1. Draw diagonal AC = 5.4 cm.
2. Draw the perpendicular bisector of AC to find its mid-point O.
3. From O, cut off OB = 2.7 cm and OD = 2.7 cm (since diagonals of a square are equal and bisect each other at right angles).
4. Join AB, BC, CD, and DA.
5. ABCD is the required square.
31. Perimeter = 24 cm.
Answer Steps:
1. Calculate side: Side = Perimeter / 4 = 24 / 4 = 6 cm.
2. Draw AB = 6 cm.
3. At A and B, construct angles of 90°.
4. Cut off AD = 6 cm and BC = 6 cm.
5. Join CD.
6. ABCD is the required square.
32. Construct a rhombus, having given one side = 4.8 cm and one angle = 75°.
Answer Steps:
1. Draw AB = 4.8 cm.
2. At A, construct an angle of 75°.
3. Cut off AD = 4.8 cm from this ray.
4. With D as centre and radius 4.8 cm, draw an arc.
5. With B as centre and radius 4.8 cm, draw an arc intersecting the previous arc at C.
6. Join DC and BC.
7. ABCD is the required rhombus.
33. Construct a regular hexagon of side (i) 2.5 cm (ii) 3.2 cm.
Answer Steps:
1. (i) For side = 2.5 cm: Draw a circle of radius 2.5 cm with centre O.
2. Take any point A on the circumference of the circle.
3. With A as centre and radius 2.5 cm, cut an arc on the circle at B.
4. Similarly, from B cut C, from C cut D, from D cut E, and from E cut F on the circumference using the same 2.5 cm radius.
5. Join AB, BC, CD, DE, EF, and FA. ABCDEF is the required regular hexagon.
6. (ii) For side = 3.2 cm: Draw a circle of radius 3.2 cm with centre O.
7. Repeat the exact same steps (2 to 5) using a compass radius of 3.2 cm to mark arcs on the circumference and join the points.
TEST YOURSELF
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) Is it possible to construct a quadrilateral with sides 5 cm, 6 cm, 7 cm, 8 cm and one of the diagonals 15 cm.
(i) Yes
(ii) No
(iii) Nothing can be said
Answer: (ii) No (In any triangle formed by the diagonal, the sum of two sides must be greater than the third side. Here, 5+6=11 which is not greater than 15, and 7+8=15 which is not greater than 15).
(b) In a regular hexagon, leading diagonal of it is twice of its side.
(i) Yes
(ii) No
(iii) Nothing can be said
Answer: (i) Yes (A regular hexagon consists of 6 equilateral triangles meeting at the center; the leading diagonal is the diameter of the circumcircle, which equals two times the side).
(c) Statement (1) : For a quadrilateral ABCD if AB = BC = CD = DA = 8 cm, then it is possible to construct this quadrilateral.
Statement (2) : It is only possible to construct this quadrilateral if each of its diagonals is greater than 8 cm.
(i) Both the statements are true.
(ii) Both the statements are false.
(iii) Statement 1 is true, and statement 2 is false.
(iv) Statement 1 is false, and statement 2 is true.
Answer: (ii) Both the statements are false.
(d) Assertion (A) : A parallelogram can be constructed if the measures of its diagonals and one side are given.
Reason (R) : It is possible to construct this parallelogram as the diagonals bisect each other.
(i) A is true, R is false.
(ii) A is false, R is true.
(iii) Both A and R are true and R is the correct reason for A.
(iv) Both A and R are true and R is the incorrect reason for A.
Answer: (iii) Both A and R are true and R is the correct reason for A.
2. Construct a quadrilateral ABCD with AB = 7 cm, BC = CD = 5 cm and ∠ABC = ∠BCD = 90°.
Answer Steps:
1. Draw BC = 5 cm.
2. At B, construct an angle of 90°.
3. Cut off AB = 7 cm on this perpendicular ray.
4. At C, construct an angle of 90°.
5. Cut off CD = 5 cm on this perpendicular ray.
6. Join AD.
7. ABCD is the required quadrilateral.
3. Construct a trapezium ABCD in which AD//BC, AB = CD = 3.6 cm, BC = 5 cm and AD = 4.5 cm.
Answer Steps:
1. Draw BC = 5 cm.
2. Mark a point E on BC such that EC = AD = 4.5 cm. (This means BE = 5 - 4.5 = 0.5 cm).
3. Construct triangle ABE with AB = 3.6 cm, BE = 0.5 cm, and AE = CD = 3.6 cm.
4. From A, draw a line parallel to BC.
5. With A as centre and radius 4.5 cm, draw an arc cutting the parallel line at D.
6. Join CD.
7. ABCD is the required trapezium.
4. Using ruler and compasses, construct a rectangle ABCD with AB = 5 cm and AD = 3.6 cm.
Answer Steps:
1. Draw AB = 5 cm.
2. At A and B, construct angles of 90°.
3. Cut off AD = 3.6 cm from the perpendicular at A and BC = 3.6 cm from the perpendicular at B.
4. Join CD.
5. ABCD is the required rectangle.
5. Using ruler and compasses only, construct the quadrilateral ABCD, having given AB = 5 cm, BC = 2.5 cm, CD = 6 cm, angle BAD = 90° and the diagonal AC = 5.5 cm.
Answer Steps:
1. Draw AB = 5 cm.
2. At A, construct an angle of 90° for the ray forming AD.
3. With A as centre and radius 5.5 cm (AC), draw an arc.
4. With B as centre and radius 2.5 cm (BC), draw an arc intersecting the previous arc at C.
5. Join BC and AC.
6. With C as centre and radius 6 cm (CD), draw an arc cutting the perpendicular 90° ray from A at point D.
7. Join CD.
8. ABCD is the required quadrilateral.
6. Using ruler and compasses only, construct a trapezium ABCD, in which the parallel sides AB and DC are 3.3 cm apart; AB = 4.5 cm, angle A = 120°, BC = 4.2 cm and angle B is obtuse. (HOTS)
Answer Steps:
1. Draw AB = 4.5 cm.
2. Draw a line parallel to AB at a perpendicular distance of 3.3 cm.
3. At A, construct an interior angle of 120° cutting the parallel line at D.
4. With B as centre and radius 4.2 cm, draw an arc cutting the parallel line at C. (To ensure angle B is obtuse, mark the intersection C that falls to the right side of point B).
5. Join AD and BC.
6. ABCD is the required trapezium.
7. Using ruler and compasses only, construct the quadrilateral ABCD, having given AB = 5 cm, BC = 2.5 cm, CD = 6 cm, ∠BAD = 90° and diagonal BD = 5.5 cm.
Answer Steps:
1. Draw AB = 5 cm.
2. At A, construct an angle of 90°.
3. With B as centre and radius 5.5 cm (BD), draw an arc cutting the perpendicular line from A at point D.
4. Join BD and AD.
5. With B as centre and radius 2.5 cm (BC), draw an arc.
6. With D as centre and radius 6 cm (CD), draw an arc intersecting the previous arc at C.
7. Join BC and DC.
8. ABCD is the required quadrilateral.
8. Using ruler and compasses only, construct a parallelogram ABCD using the following data: AB = 6 cm, AD = 3 cm and ∠DAB = 45°. If the bisector of ∠DAB meets DC at P, prove that ∠APB is a right angle.
Answer Steps:
1. Draw AB = 6 cm.
2. At A, construct an angle of 45° and cut off AD = 3 cm.
3. With D as centre and radius 6 cm, draw an arc. With B as centre and radius 3 cm, draw an arc to intersect at C.
4. Join DC and BC. ABCD is the parallelogram.
5. Construct the angle bisector of ∠DAB. Let it meet DC at P. Join PB.
6. Proof: Since AP bisects ∠DAB, ∠DAP = ∠PAB = 22.5°.
7. Since AB // DC, alternate interior angles are equal: ∠DPA = ∠PAB = 22.5°.
8. In ΔDAP, ∠DAP = ∠DPA = 22.5°, hence it is an isosceles triangle with AD = DP.
9. Given AD = 3 cm, so DP = 3 cm.
10. Since DC = AB = 6 cm, PC = DC - DP = 6 - 3 = 3 cm.
11. In parallelogram ABCD, consecutive angles add to 180°, so opposite angle ∠C = ∠DAB = 45° and BC = AD = 3 cm.
12. In ΔPCB, PC = BC = 3 cm, so it is an isosceles triangle.
13. ∠CPB = ∠CBP = (180° - ∠C) / 2 = (180° - 45°) / 2 = 135° / 2 = 67.5°.
14. DCP is a straight line, so ∠DPA + ∠APB + ∠CPB = 180°.
15. 22.5° + ∠APB + 67.5° = 180°.
16. 90° + ∠APB = 180° which gives ∠APB = 90°. (Proved).
9. The perpendicular distances between the pair of opposite sides of a parallelogram are 3 cm and 4 cm, and one of its angles measures 60°. Using ruler and compasses only, construct the parallelogram.
Answer Steps:
1. Draw a base line L1.
2. Draw a line L2 parallel to L1 at a perpendicular distance of 3 cm.
3. Take a point A on L1. Construct an angle of 60° at A.
4. Let the arm of this angle intersect L2 at D. The line segment AD is one side.
5. Draw a perpendicular to AD from A, mark a point at exactly 4 cm distance.
6. Draw a line L3 perpendicular to this new segment, which makes L3 parallel to AD at a distance of 4 cm.
7. Let L3 intersect L1 at B and L2 at C.
8. ABCD is the required parallelogram.
10. Draw parallelogram ABCD with the following data : (HOTS)
AB = 6 cm, AD = 5 cm and ∠DAB = 45°.
Let AC and DB meet in O and let E be the mid-point of BC. Join OE. Prove that :
(i) OE // AB
(ii) OE = 1/2 AB.
Answer Steps:
1. Draw AB = 6 cm.
2. At A, construct an angle of 45° and cut off AD = 5 cm.
3. With D as centre and radius 6 cm, draw an arc. With B as centre and radius 5 cm, draw an arc intersecting at C.
4. Join DC and BC to complete parallelogram ABCD.
5. Join diagonals AC and BD to meet at O. Mark the mid-point E of BC. Join OE.
6. Proof for (i): In a parallelogram, diagonals bisect each other. Therefore, O is the exact mid-point of diagonal AC.
7. In ΔABC, O is the mid-point of AC and E is the given mid-point of BC.
8. By the Mid-point Theorem, the line segment joining the mid-points of two sides of a triangle is parallel to the third side.
9. Therefore, OE // AB. (Proved).
10. Proof for (ii): By the same Mid-point Theorem, the length of this joined segment is half of the third side.
11. Therefore, OE = 1/2 AB. (Proved).
11. Using ruler and compasses only, construct a rectangle each of whose diagonals measure 6 cm and the diagonals intersect at an angle of 45°.
Answer Steps:
1. Draw a line segment AC = 6 cm as the first diagonal.
2. Draw the perpendicular bisector of AC to find its mid-point O. (AO = OC = 3 cm).
3. At O, construct an angle of 45° with AC and extend this line on both sides of O.
4. Since the diagonals of a rectangle are equal in length and bisect each other, cut off OB = 3 cm and OD = 3 cm on this new line.
5. Join AB, BC, CD, and DA.
6. ABCD is the required rectangle.
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Quick Review Flashcards - Click to flip and test your knowledge!
Question
What should always be drawn before starting the actual construction of a quadrilateral?
Answer
A rough free-hand sketch.
Question
To 'construct' a quadrilateral essentially means to find or locate its four _____.
Answer
vertices
Question
When constructing a quadrilateral with four sides and one angle given, what is the first line segment to be drawn?
Answer
The base side adjacent to the given angle.
Question
If quadrilateral $ABCD$ has $AB = 3.5\text{ cm}$, $BC = 4.0\text{ cm}$, and $\angle B = 45^{\circ}$, which point is located by cutting an arc of $3.5\text{ cm}$ from point $B$?
Answer
Vertex $A$
Question
When constructing a quadrilateral with three sides and two consecutive angles, which side is typically drawn first?
Answer
The side included between the two given angles.
Question
In the construction of a quadrilateral with four sides and one diagonal, the first step is to construct a _____ using three of the given dimensions.
Answer
triangle
Question
If quadrilateral $ABCD$ has sides $AB$, $BC$, $CD$, $DA$ and diagonal $AC$ given, which vertex is found by the intersection of arcs from $A$ and $C$ after $\Delta ABC$ is built?
Answer
Vertex $D$
Question
What is the first procedural step to construct a quadrilateral when three sides and two diagonals are known?
Answer
Construct a triangle using one side and both diagonals (or two sides and one diagonal).
Question
In a parallelogram, what is the relationship between opposite sides regarding their length?
Answer
Opposite sides are equal.
Question
If parallelogram $ABCD$ has $AB = 3.0\text{ cm}$ and $BC = 4.0\text{ cm}$, what is the length of side $AD$?
Answer
$4.0\text{ cm}$
Question
To construct a parallelogram when two consecutive sides and the included angle are given, one first constructs a _____.
Answer
triangle
Question
What key geometric property of parallelogram diagonals is used for construction when only one side and both diagonals are given?
Answer
The diagonals bisect each other.
Question
In a parallelogram with diagonals $AC$ and $BD$ intersecting at $O$, if $BD = 5.0\text{ cm}$, what is the length of segment $OB$?
Answer
$2.5\text{ cm}$
Question
When constructing a parallelogram from its diagonals, what is the purpose of 'producing' segment $BO$ to $D$ such that $OD = OB$?
Answer
To locate the fourth vertex while ensuring the diagonals bisect each other.
Question
If a parallelogram must be constructed with two diagonals and an included angle, what is the first step?
Answer
Draw one diagonal and locate its mid-point $O$.
Question
Why might you start a parallelogram construction with diagonal $BD = 4.5\text{ cm}$ instead of $AC = 5.4\text{ cm}$ if the included angle is given?
Answer
If half of the other diagonal ($2.25\text{ cm}$) cannot be accurately measured on a standard scale.
Question
In the construction of a parallelogram using height, what is drawn through the height point $E$ to be parallel to the base $BC$?
Answer
A perpendicular line to the altitude $CP$.
Question
To construct a trapezium $ABCD$ where $AD \parallel BC$, what is the first segment usually drawn?
Answer
The base $BC$.
Question
In a trapezium construction with four sides given, why is a point $E$ marked on $BC$ such that $BE = AD$?
Answer
To create a triangle $DEC$ where $DE$ is equal to side $AB$.
Question
What is the measure of every interior angle in a rectangle?
Answer
$90^{\circ}$
Question
To construct a rectangle with adjacent sides $AB = 3\text{ cm}$ and $BC = 5\text{ cm}$, what type of triangle is constructed first?
Answer
A right-angled triangle.
Question
When only one side and one diagonal of a rectangle are given, the construction is completed by drawing two _____ triangles.
Answer
right-angled
Question
What is the defining property of the diagonals of a rhombus regarding their intersection?
Answer
They bisect each other at right angles ($90^{\circ}$).
Question
In constructing a rhombus from diagonals $AC$ and $BD$, what must be drawn to diagonal $AC$ to find the line for $BD$?
Answer
A perpendicular bisector.
Question
If a rhombus has diagonal $AC = 6.0\text{ cm}$ and $BD = 4.6\text{ cm}$, how far from the intersection point $O$ are the vertices $B$ and $D$ located?
Answer
$2.3\text{ cm}$
Question
The construction method for a square given its diagonal is identical to the method used for a _____.
Answer
rhombus
Question
What is the value of each interior angle in a regular hexagon?
Answer
$120^{\circ}$
Question
In Method 1 of constructing a regular hexagon, if side $AB = 3.0\text{ cm}$, what is the measure of $\angle PAB$?
Answer
$120^{\circ}$
Question
Method 2 for constructing a regular hexagon relies on the fact that the side length is equal to the _____ of its circumcircle.
Answer
radius
Question
When using Method 2 for a regular hexagon of side $3.0\text{ cm}$, how many times is the radius length marked along the circumference?
Answer
Six times.
Question
In Method 3 for a regular hexagon, what is the angle subtended by each side at the centre of the circumcircle?
Answer
$\frac{360^{\circ}}{6} = 60^{\circ}$
Question
Constructing an equilateral triangle $AOB$ inside a circle where $O$ is the centre is a step in constructing which polygon?
Answer
A regular hexagon.
Question
Can a quadrilateral be constructed if only four sides are known?
Answer
No, at least one more part (like an angle or diagonal) is required.
Question
Is it possible to construct a unique parallelogram if only the lengths of the two diagonals are given?
Answer
No, the angle between the diagonals is also required.
Question
True or False: A rhombus can be constructed if only the lengths of its two diagonals are known.
Answer
True
Question
If the diagonals of a quadrilateral are equal and bisect each other at right angles, the figure is a _____.
Answer
square
Question
In the construction of a parallelogram given two adjacent sides and an altitude, the altitude is the distance between _____ sides.
Answer
parallel
Question
To find the mid-point of a line segment using only a compass and ruler, one constructs a _____.
Answer
perpendicular bisector
Question
If a regular hexagon's leading diagonal is twice the length of its side, can it be constructed from the diagonal alone?
Answer
Yes.
Question
In a regular hexagon, opposite sides are always _____.
Answer
parallel
Question
When constructing a regular hexagon by dividing a circle, what is the relationship between the chord length and the radius?
Answer
They are equal.
Question
For a quadrilateral $ABCD$ to be constructed, the sum of any three angles must be less than _____.
Answer
$360^{\circ}$
Question
A trapezium can be constructed if four _____ are given.
Answer
sides
Question
How many degrees are in the sum of the interior angles of any quadrilateral?
Answer
$360^{\circ}$
Question
In a parallelogram $ABCD$, if $\angle A = 75^{\circ}$, what is the measure of the consecutive angle $\angle B$?
Answer
$105^{\circ}$
Question
To construct a rectangle $ABCD$ with $AB = 6.0\text{ cm}$ and $AD = 5.0\text{ cm}$, what angle must be constructed at vertex $A$?
Answer
$90^{\circ}$
Question
Which specific tool is used to draw arcs of a specific radius in geometric constructions?
Answer
A compass.
Question
In the construction of a parallelogram using diagonals $AC$ and $BD$ and an acute angle between them, if the angle is $60^{\circ}$, which angle at the intersection $O$ is $60^{\circ}$?
Answer
$\angle DOC$ (or $\angle AOB$).
Question
In the construction of a trapezium $ABCD$ where $AD \parallel BC$, if $AD = 3\text{ cm}$ and $BC = 5\text{ cm}$, what length is $EC$ if $BE = AD$?
Answer
$2\text{ cm}$
Question
To locate vertex $D$ in a quadrilateral construction where $CD = 5\text{ cm}$ and $DA = 4\text{ cm}$ are given, from which two points should arcs be drawn?
Answer
Points $C$ and $A$.
Question
When constructing a regular hexagon $ABCDEF$, why is it possible to use the radius of the circle to mark all vertices?
Answer
Because the hexagon is composed of six equilateral triangles meeting at the centre.
Question
What is the measure of the exterior angle of a regular hexagon?
Answer
$60^{\circ}$
Question
If the sides of a quadrilateral are $5\text{ cm}$, $6\text{ cm}$, $7\text{ cm}$, $8\text{ cm}$ and a diagonal is $15\text{ cm}$, why is construction impossible?
Answer
Because the diagonal is longer than the sum of two sides (violating the triangle inequality).
Question
In a rhombus $ABCD$, if diagonal $AC = 6.3\text{ cm}$ and $BD = 5.8\text{ cm}$, what is the distance from the intersection $O$ to vertex $A$?
Answer
$3.15\text{ cm}$
Question
Which property ensures that a rectangle can be constructed with just two adjacent sides given?
Answer
All angles are $90^{\circ}$ and opposite sides are equal.
Question
If the perimeter of a square is $24\text{ cm}$, what side length is used for construction?
Answer
$6\text{ cm}$
Question
To construct a parallelogram when two adjacent sides and the height corresponding to one side are given, which side is drawn first?
Answer
The side to which the height corresponds.
Question
In the construction of a regular hexagon by Method 1, after drawing side $AB$ and the $120^{\circ}$ angles at $A$ and $B$, which points are located next?
Answer
Points $F$ and $C$.
Question
A quadrilateral can be constructed if its four sides and one _____ are known.
Answer
angle (or diagonal)
Question
In a parallelogram, if $\angle DAB = 45^{\circ}$, what is the measure of the opposite angle $\angle BCD$?
Answer
$45^{\circ}$