Proving and Applying Similar Triangles
Every question with a fully worked solution.
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Theory
Similar triangles have equal matching angles and matching sides in the same ratio. This Year 9 maths page shows how to prove that two triangles are similar with a reason for each step, match their sides, and find unknown lengths, including parallel lines inside a triangle, triangles that cross at a point, and real problems with shadows, mirrors and rivers. Formal geometric proof is taken further in Year 10.
Two triangles are similar if one is an enlargement of the other. Their matching angles are equal and their matching sides are in the same ratio, the scale factor. We write \(\triangle PQR \sim \triangle XYZ\) (some books use \(|\!|\!|\)), with the vertices in matching order.
Two common ways to see that triangles are similar: all three pairs of sides are in the same ratio, or two pairs of angles are equal (then the third pair is equal too, by the angle sum).
Equal angles often come from a common angle shared by both triangles, a pair of right angles, vertically opposite angles, or corresponding and alternate angles on parallel lines.
Matching sides are opposite matching angles. The shortest side matches the shortest side and the longest matches the longest.
Similar triangles let you measure things you cannot reach, such as the height of a tree or the width of a river, from lengths you can measure.
If \(\triangle PQR \sim \triangle XYZ\):
To find an unknown \(x\), write it over its matching side and set it equal to another known pair:
How to find an unknown side in similar triangles
- Show the triangles are similar: give two pairs of equal angles with reasons, or three sides in the same ratio.
- Separate overlapping triangles. Sketch them side by side, the same way round, using whole sides.
- Match the sides: each side is opposite a matching angle.
- Find the scale factor from a known pair, or write the ratio equation with the unknown on top.
- Solve and check that the bigger triangle has the bigger sides.
Setting out a similarity proof
- Name the two triangles: "In \(\triangle ADE\) and \(\triangle ABC\)".
- List the equal angles (or equal ratios of sides), one per line, each with its reason: common angle, vertically opposite, corresponding or alternate angles with the parallel lines named, or given right angles.
- Conclude with the statement in matching order and the test in brackets: \(\therefore\ \triangle ADE \sim \triangle ABC\) (AA).
- Then use it: matching sides are in the same ratio, so write the ratio equation for the length you want.
Real problems
- Shadows: height matches height, shadow matches shadow (same sun angle).
- Mirror on the ground: the angles at the mirror are equal, so eye height \(\div\) distance to the mirror matches object height \(\div\) its distance to the mirror.
- Lining up two tops: a short pole and a tall tower seen from one point share the angle at that point. Use the whole distance to the tower.
Match shortest with shortest, middle with middle, longest with longest, and divide.
| \(\dfrac{9}{6}\) | \(=\) | \(1.5\) |
| \(\dfrac{12}{8}\) | \(=\) | \(1.5\) |
| \(\dfrac{13.5}{9}\) | \(=\) | \(1.5\) |
All three ratios are equal, so the triangles are similar with scale factor \(1.5\).
\(\angle K\) is opposite the \(6\) cm side \(LM\). The matching side is \(YZ = 9\) cm, which is opposite \(\angle X\). So \(\angle K\) matches \(\boldsymbol{\angle X}\).
(i) Prove that \(\triangle ADE \sim \triangle ABC\).
(ii) Find \(BC\).
(iii) \(AC = 17.5\) cm. Find \(AE\).
(i) In \(\triangle ADE\) and \(\triangle ABC\):
| \(\angle DAE\) | \(=\) | \(\angle BAC\) |
| \(\text{(common angle)}\) | ||
| \(\angle ADE\) | \(=\) | \(\angle ABC\) |
| \(\text{(corresponding, } DE \parallel BC)\) |
\(\therefore\ \triangle ADE \sim \triangle ABC\) (AA).
(ii) and (iii) The large triangle uses the whole side \(AB = 9 + 6 = 15\) cm.
| \(k\) | \(=\) | \(\dfrac{AB}{AD} = \dfrac{15}{9} = \dfrac{5}{3}\) |
| \(BC\) | \(=\) | \(12 \times \dfrac{5}{3} = \mathbf{20}\text{ cm}\) |
| \(AE\) | \(=\) | \(17.5 \div \dfrac{5}{3} = \mathbf{10.5}\text{ cm}\) |
Keep \(k\) as the fraction \(\dfrac{5}{3}\) rather than \(1.67\), so the answers stay exact.
Both triangles have a right angle, and the sun's rays meet the ground at the same angle. So the triangles are similar: height matches height and shadow matches shadow.
| \(\dfrac{h}{1.8}\) | \(=\) | \(\dfrac{14}{2.4}\) |
| \(h\) | \(=\) | \(\dfrac{1.8 \times 14}{2.4}\) |
| \(\) | \(=\) | \(\mathbf{10.5}\text{ m}\) |
Writing the ratios as unknown over its match puts \(h\) on top, so one multiplication finishes it.
\(\angle TCB = \angle ECD\) (vertically opposite) and \(\angle TBC = \angle EDC = 90^\circ\), so \(\triangle TBC \sim \triangle EDC\). Match the sides: \(BT \leftrightarrow DE\) and \(BC \leftrightarrow DC\).
| \(\dfrac{w}{20}\) | \(=\) | \(\dfrac{36}{15}\) |
| \(w\) | \(=\) | \(20 \times \dfrac{36}{15}\) |
| \(\) | \(=\) | \(\mathbf{48}\text{ m}\) |
The side next to the shared corner \(C\) on the bank matches the side next to \(C\) on the other triangle.
Common pitfalls
Frequently asked questions
How do you know if two triangles are similar?
Two triangles are similar if two pairs of angles are equal, or if all three pairs of sides are in the same ratio. Equal angles often come from a shared angle, right angles or parallel lines.
How do you find the missing side of a similar triangle?
Match the sides, find the scale factor from a known pair, then multiply or divide. Or write the unknown over its matching side equal to another known ratio and solve.
Which sides match in similar triangles?
Sides opposite equal angles match. The shortest side matches the shortest, and the longest matches the longest. In a similarity statement, the letters in the same positions match.
How do you use shadows to find the height of a tree?
At the same moment the sun makes the same angle for every object, so the triangles are similar. Tree height \(\div\) tree shadow \(=\) your height \(\div\) your shadow.
What is the difference between similar and congruent triangles?
Similar triangles have the same shape but can be different sizes. Congruent triangles have the same shape and the same size, so the scale factor is \(1\).