Use a Matched Length Ratio

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Two similar triangles have corresponding sides 6 and 12. A second side on the smaller triangle is 8. Find its corresponding larger side. Candidate answers: (A) 15 cm (B) 16 cm (C) 17 cm (D) 32 cm.
• Independent check inputs: Two similar triangles have corresponding sides 9 and 18. A second side on the smaller triangle is 12. Find its corresponding larger side. Candidate answers: (A) 23 cm (B) 24 cm (C) 25 cm (D) 48 cm.

Learn the idea

Similarity preserves angle measures and corresponding length ratios. Right triangles with the same acute angle are similar, explaining why sine, cosine and tangent depend on an angle rather than the triangle’s overall size. Corresponding side lengths of similar triangles share one scale factor. Find that factor from a known corresponding pair, then apply it to the requested matched side.

Here is the method on the supplied teaching case. The larger-to-smaller length ratio is 12/6=2. Multiply the corresponding smaller side 8 by 2 to obtain 16. Match corresponding sides before writing the ratio.

Correspondence must be established before forming a ratio. Keep larger-to-smaller or smaller-to-larger order consistent across pairs; mixing those orders creates an incorrect reciprocal scale factor.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Two similar triangles have corresponding sides 27 and 54. A second side on the smaller triangle is 36. Find its corresponding larger side. Candidate answers: (A) 144 cm (B) 71 cm (C) 72 cm (D) 73 cm. Show the mathematical reason, not only the selected answer. Expected result: 72 cm. The larger-to-smaller length ratio is 54/27=2. Multiply the corresponding smaller side 36 by 2 to obtain 72. Match corresponding sides before writing the ratio.

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