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MathidiaMathematics Academy
Similarity and congruence
Lesson 45 of 75
Review

Similarity problems

Objective. Combine ratio, algebra and geometry in multi-step solutions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Similarity problems combine ratio, algebra and geometric reasoning across multiple steps, often requiring an unknown length to be found using an equation built from the equal ratios of corresponding sides in similar shapes. The typical method is to set up a proportion equation between two pairs of matching sides, then solve the resulting equation using algebraic techniques such as cross-multiplication. For example, if similar triangles give the proportion x/6 = 9/12, cross-multiplying gives 12x = 54, so x = 4.5. These problems often embed similarity inside a wider context, such as shadows, maps or nested triangles, requiring careful identification of which sides correspond before any algebra begins.

Worked example

Two similar triangles have corresponding sides where one triangle's side of length x corresponds to 8 cm, matched against a known ratio of 5 cm to 20 cm elsewhere in the similar triangles. Find x.

Diagram of a triangle
  1. 1Set up the proportion using corresponding sides: x/8 = 5/20.
  2. 2Cross-multiply: 20x = 8 × 5.
  3. 3Calculate the right-hand side: 20x = 40.
  4. 4Divide both sides by 20: x = 2.

Answer: x = 2 cm

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Common mistake. Pupils sometimes match sides incorrectly between the two similar shapes, pairing a short side in one triangle with a long side in the other, which produces an equation with the wrong proportion. Always match corresponding sides using equal angles as the guide before writing the ratio.

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