Revision notes
Spot similar triangles, use the scale factor to find missing sides, and measure tall things with shadows.
Musa wants to know the height of the big mango tree in his compound. With a metre stick, a tape and the afternoon sun, he can find it without leaving the ground — thanks to similar triangles.
Two triangles are similar when one is an enlargement of the other: the same shape, but not always the same size.
In similar triangles, the matching angles are equal, and every side is multiplied by the same number, the scale factor.
Enlarging a triangle changes its sides but never its angles.
🔑 Key idea: Similar triangles: equal angles, and sides in the same ratio.
First pair up the matching sides. Divide a pair of known matching sides to get the scale factor.
If AB = 4 cm matches PQ = 12 cm, the scale factor is 12 ÷ 4 = 3. So if BC = 5 cm, then QR = 5 × 3 = 15 cm.
Going from the big triangle to the small one, divide by the scale factor instead.
To test whether two triangles are similar, divide each pair of matching sides: all the answers must be the same.
🔑 Key idea: Scale factor = big side ÷ matching small side. Multiply to go up, divide to go down.
At the same moment, the sun's rays hit every upright object at the same angle, so each object and its shadow make similar triangles.
A 2 m stick casting a 3 m shadow tells you that a tree with a 12 m shadow is 12 ÷ 3 = 4 times as tall: 2 × 4 = 8 m.
🔑 Key idea: Height ÷ shadow is the same for everything at the same time of day.
Triangles ABC and PQR are similar, with AB matching PQ and BC matching QR. PQ = 21 cm, QR = 9 cm and AB = 7 cm. Find BC.
Step 1: Scale factor
PQ ÷ AB = 21 ÷ 7 = 3
Step 2: Divide the matching side
BC = 9 ÷ 3 = 3 cm
Answer: 3 cm
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