Covers Lessons 15–20: applying congruence, similar figures, finding missing sides in similar figures, multi-step geometric reasoning, constructions (bisecting angles and lines) and a synthesis of the unit.
Question 9 3 marks
Two rectangles are similar. The smaller has width 6 cm; the matching width on the larger is 18 cm.
(a) Find the scale factor from the smaller rectangle to the larger. 1 mark
(b) The smaller rectangle has length 5 cm. Find the matching length on the larger rectangle. 1 mark
(c) What scale factor would make the two rectangles congruent instead of similar? 1 mark
(a) Scale factor = larger ÷ smaller = 18 ÷ 6 = 3 [1]
(b) Matching length = 5 × 3 = 15 cm [1]
(c) A scale factor of 1 (the figures would then be the same size, i.e. congruent) [1]
Question 10 4 marks
(a) Triangles ABC and DEF are similar. Side AB = 4 cm matches DE = 10 cm, and BC = 6 cm. Find the matching side EF. 2 marks
(b) A person 1.8 m tall casts a 3 m shadow. At the same time a tree casts a 12 m shadow. Use similar triangles to find the height of the tree. 2 marks
(a) Scale factor = DE ÷ AB = 10 ÷ 4 = 2.5; EF = BC × 2.5 = 6 × 2.5 = 15 cm [2]
(b) height ÷ shadow is constant: 1.8 ÷ 3 = 0.6; tree height = 0.6 × 12 = 7.2 m [2]
Question 11 3 marks
(a) Describe how to use a compass and straightedge to bisect an angle. 1 mark
(b) An angle on a straight line next to a 130° angle is the base of a triangle. The triangle's other two angles are this base angle and 60°. Find the third angle of the triangle. 2 marks
(a) Place the compass point on the vertex and draw an arc crossing both arms. From each crossing point, draw two equal arcs that meet inside the angle. Draw a straight line from the vertex through that meeting point, this line bisects the angle into two equal halves [1].
(b) Base angle on the straight line = 180 − 130 = 50° [1]. Third angle of the triangle = 180 − 50 − 60 = 70° [1].