Triangles

Two figures with the same shape but not necessarily the same size are called similar figures. All circles are similar, all squares are similar, and a photograph and its enlargement are similar. In this chapter you will learn that two polygons with the same number of sides are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. Similarity is the idea that lets us find heights of tall objects such as towers and trees from their shadows. The central result is the Basic Proportionality Theorem (Thales theorem): if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Its converse lets us test whether a line is parallel to a side. You will then study the criteria for similarity of triangles, AAA (or AA), SSS and SAS, and use them to solve problems on shadows, trapeziums and lamp posts.

Detailed Worksheet: Triangles Section A - Definitions (10 marks) 1. What are similar figures? Are all congruent figures similar? Are all similar figures congruent? (2 marks) 2. State the two conditions for two polygons with the same number of sides to be similar. (2 marks) 3. State the Basic Proportionality Theorem and its converse. (2 marks) 4. State the AAA, SSS and SAS similarity criteria for triangles. (2 marks) 5. Explain why the AA criterion is enough to prove two triangles similar. (2 marks) Section B - Calculations and Applications (15 marks) 6. In triangle ABC, DE is parallel to BC, with D on AB and E on AC. If AD = 1.5 cm, DB = 3 cm and AE = 1 cm, find EC. (3 marks) 7. In triangle ABC, DE is parallel to BC. If AD = 4x - 3, DB = 3x - 1, AE = 8x - 7 and EC = 5x - 3, find the value of x. (3 marks) 8. E and F are points on the sides PQ and PR of triangle PQR. For each case, state whether EF is parallel to QR: (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm, FR = 2.4 cm (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm, RF = 9 cm. (3 marks) 9. A vertical pole of length 6 m casts a shadow 4 m long on the ground, and at the same time a tower casts a shadow 28 m long. Find the height of the tower. (3 marks) 10. Triangle ABC is similar to triangle DEF with AB = 3 cm, BC = 4 cm, CA = 5 cm and DE = 6 cm. Find EF, FD and the ratio of the perimeters of the two triangles. (3 marks) Section C - Diagrams (10 marks) 11. Draw a triangle ABC and a line DE parallel to BC meeting AB at D and AC at E. Join BE and CD and draw the perpendiculars needed to prove the Basic Proportionality Theorem using areas of triangles. Write the proof briefly. (4 marks) 12. Draw a trapezium ABCD with AB parallel to DC, whose diagonals intersect at O. Show that AO/BO = CO/DO, naming the similarity criterion used. (3 marks) 13. Draw the figure of a girl of height 90 cm walking away from the base of a lamp post 3.6 m high, showing the lamp, the girl and her shadow. Mark the two similar triangles. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. Explain which triangles are similar and why. (5 marks) 15. Using the Basic Proportionality Theorem, prove that a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side. Name the familiar theorem obtained, and explain how it is a special case of the BPT. (5 marks) 16. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that triangle ABC is similar to triangle PQR. Explain why showing just two pairs of proportional sides would not be enough. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw a neat figure for every question, mark the given parts, and name the similarity criterion or theorem used at each step.