Similar Triangle Sides: Step-by-Step Guide
When two triangles are similar, their matching sides are in proportion. That means the ratio of one pair of matching sides equals the ratio of every other pair. To find a missing side, set up a proportion (two equal fractions) and solve it by cross-multiplying.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Match the sides. Use the similarity statement (like triangle ABC ~ triangle DEF) to pair up the sides: AB with DE, BC with EF, and CA with FD.\[\triangle ABC \sim \triangle DEF \Rightarrow AB \leftrightarrow DE,\; BC \leftrightarrow EF,\; CA \leftrightarrow FD\]
- Find the scale factor. Divide one known side by its matching side. This ratio is the same for every pair of matching sides.\[k = \frac{\text{side in big triangle}}{\text{corresponding side in small triangle}}\]
- Set up a proportion. Write two equal fractions: the unknown side over its matching side equals the scale factor.\[\frac{x}{\text{known counterpart}} = k\]
- Cross-multiply and solve. Multiply across the equals sign to remove the fractions, then solve for the unknown.\[x = k \cdot \text{known counterpart}\]
- Check with another pair. Make sure the ratio of another pair of matching sides gives the same scale factor.\[\frac{x}{\text{counterpart}} \stackrel{?}{=} k\]
Worked example
Triangle ABC ~ Triangle DEF. AB = 6, BC = 8, DE = 9. Find EF.
Step 1: Match the sides: AB goes with DE, and BC goes with EF.
Step 2: Find the scale factor using AB and DE: DE / AB = 9 / 6 = 3/2.
Step 3: Set up the proportion: EF / BC = 3/2, so EF / 8 = 3/2.
Step 4: Cross-multiply: 2 x EF = 3 x 8 = 24.
Step 5: Solve: EF = 24 / 2 = 12.
Check: EF / BC = 12 / 8 = 3/2, which is the same as DE / AB. The ratios match, so EF = 12.
Tips and common mistakes
- Always match the sides in the right order. Read the similarity statement carefully.
- The scale factor is the same for every pair of matching sides. Use the pair that is easiest to work with.
- If the scale factor is bigger than 1, the second triangle is bigger. If it is less than 1, the second triangle is smaller.
- Cross-multiplying: if a/b = c/d, then a x d = b x c.
- If you are not sure, draw both triangles and label the matching corners the same way.
- The perimeters of similar triangles have the same ratio as the sides.
- The areas have a ratio equal to the scale factor squared (k x k), not just k.
- Check your answer by testing another pair of sides.