CONSTRUCTIONS (A) Main Concepts and Results • To bisect a given angle, • To draw the perpendicular bisector of a line segment, • To construct angles of 15°, 30°, 45°, 60°, 90°, etc. • To construct a triangle given its base, a base angle and the sum of other two sides, • To construct a triangle given its base, a base angle and the difference of other two sides, • To construct a triangle given its perimeter and the two base angles • Geometrical construction means using only a ruler and a pair of compasses as geometrical instruments. (B) Multiple Choice Questions Sample Question 1: With the help of a ruler and a compass, it is possible to construct an angle of : (A) 35° (B) 40° (C) 37.5° (D) 47.5° Solution : Answer (C) Sample Question 2: The construction of a triangle ABC in which AB = 4 cm, ∠A = 60° is not possible when difference of BC and AC is equal to: (A) 3.5 cm (B) 4.5 cm (C) 3 cm (D) 2.5 cm Solution : Answer (B) CONSTRUCTIONS EXERCISE 11.1 1. With the help of a ruler and a compass it is not possible to construct an angle of : (A) 37.5° (B) 40° (C) 22.5° (D) 67.5° 2. The construction of a triangle ABC, given that BC = 6 cm, ∠ B = 45° is not possible when difference of AB and AC is equal to: (A) 6.9 cm (B) 5.2 cm (C) 5.0 cm (D)4.0 cm 3. The construction of a triangle ABC, given that BC = 3 cm, ∠C = 60° is possible when difference of AB and AC is equal to : (A) 3.2 cm (B) 3.1 cm (C) 3 cm (D) 2.8 cm (C) Short Answer Questions with Reasoning Write True or False and give reasons for your answer. Sample Question 1 : An angle of 67.5° can be constructed. 135° 1 Solution : True. As 67.5° = (90°+ 45 ) . =° 22 EXERCISE 11.2 Write True or False in each of the following. Give reasons for your answer: 1. An angle of 52.5° can be constructed. 2. An angle of 42.5° can be constructed. 3. A triangle ABC can be constructed in which AB = 5 cm, ∠A = 45° and BC + AC = 5 cm. 4. A triangle ABC can be constructed in which BC = 6 cm, ∠C = 30° and AC – AB = 4 cm. 5. A triangle ABC can be constructed in which ∠ B = 105°, ∠C = 90° and AB + BC + AC = 10 cm. 6. A triangle ABC can be constructed in which ∠ B = 60°, ∠C = 45° and AB + BC + AC = 12 cm. (D) Short Answer Questions Sample Question 1 : Construct a triangle ABC in which BC = 7.5 cm, ∠B = 45° and AB – AC = 4 cm. Solution : See Mathematics Textbook for Class IX. EXEMPLAR PROBLEMS EXERCISE 11.3 1. Draw an angle of 110° with the help of a protractor and bisect it. Measure each angle. 2. Draw a line segment AB of 4 cm in length. Draw a line perpendicular to AB through A and B, respectively. Are these lines parallel? 3. Draw an angle of 80° with the help of a protractor. Then construct angles of (i) 40° (ii)160° and (iii) 120°. 4. Construct a triangle whose sides are 3.6 cm, 3.0 cm and 4.8 cm. Bisect the smallest angle and measure each part. 5. Construct a triangle ABC in which BC = 5 cm, ∠B = 60° and AC + AB = 7.5 cm. 6. Construct a square of side 3 cm. 7. Construct a rectangle whose adjacent sides are of lengths 5 cm and 3.5 cm. 8. Construct a rhombus whose side is of length 3.4 cm and one of its angles is 45°. (E) Long Answer Questions Sample Question 1 : Construct an equilateral triangle if its altitude is 6 cm. Give justification for your construction. Solution : Draw a line XY. Take any point D on this line. Construct perpendicular PD on XY. Cut a line segment AD from D equal to 6 cm. Make angles equal to 30° at A on both sides of AD, say ∠CAD and ∠BAD where B and C lie on XY. Then ABC is the required triangle. Justification Since ∠A = 30° + 30° = 60° and AD ⊥BC, Δ ABC is an equilateral triangle with altitude AD = 6 cm. Fig. 11.1 CONSTRUCTIONS EXERCISE 11.4 Construct each of the following and give justification : 1. A triangle if its perimeter is 10.4 cm and two angles are 45° and 120°. 2. A triangle PQR given that QR = 3cm, ∠ PQR = 45° and QP – PR = 2 cm. 3. A right triangle when one side is 3.5 cm and sum of other sides and the hypotenuse is 5.5 cm. 4. An equilateral triangle if its altitude is 3.2 cm. 5. A rhombus whose diagonals are 4 cm and 6 cm in lengths.