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Grade 10 · unit review

Review: Geometry: Similarity and Right-Triangle Ratios

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.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Two similar triangles have corresponding sides 21 and 42. A second side on the smaller triangle is 28. Find its corresponding larger side.
Need a hint?
  • Corresponding side lengths of similar triangles share one scale factor.
  • Correspondence must be established before forming a ratio.
2. In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 14 cm. Find its opposite side.
Need a hint?
  • For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length.
  • The opposite side changes when the reference angle changes.
3. For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 35 cm. Find the adjacent leg.
Need a hint?
  • Cosine compares the leg touching an acute reference angle with the hypotenuse.
  • The hypotenuse also touches the acute angle but is not the adjacent leg in this definition.
4. For an acute angle θ, tan θ=3/4 and its adjacent leg is 28 cm. Find its opposite leg.
Need a hint?
  • Tangent is the ratio of the opposite leg to the adjacent leg for an acute angle.
  • Keep the reference angle fixed while identifying the legs.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. 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. 2. In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 18 cm. Find its opposite side. Candidate answers: (A) 10 cm (B) 18 cm (C) 8 cm (D) 9 cm. 3. For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 45 cm. Find the adjacent leg. Candidate answers: (A) 35 cm (B) 36 cm (C) 37 cm (D) 72 cm. 4. For an acute angle θ, tan θ=3/4 and its adjacent leg is 36 cm. Find its opposite leg. Candidate answers: (A) 26 cm (B) 27 cm (C) 28 cm (D) 54 cm.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 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. 2. 9 cm. Sine is opposite/hypotenuse. If o/18=1/2, then o=18/2=9 cm. The side opposite θ depends on which acute angle was chosen. 3. 36 cm. Cosine is adjacent leg/hypotenuse. Multiply 45 by 4/5 to get 36. The hypotenuse is not counted as the adjacent leg in this ratio. 4. 27 cm. Tangent is opposite/adjacent. Solve o/36=3/4, giving o=36×3/4=27 cm. No hypotenuse is used in this ratio.

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