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Grade 10 · lesson 3 of 4

Use Adjacent Leg over Hypotenuse

Cosine compares the leg touching an acute reference angle with the hypotenuse.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 10 cm. Find the adjacent leg. Candidate answers: (A) 16 cm (B) 7 cm (C) 8 cm (D) 9 cm.
  • Independent check inputs: For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 15 cm. Find the adjacent leg. Candidate answers: (A) 11 cm (B) 12 cm (C) 13 cm (D) 24 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. Cosine compares the leg touching an acute reference angle with the hypotenuse. A supplied cosine ratio and hypotenuse determine that adjacent leg by multiplication.

Here is the method on the supplied teaching case. Cosine is adjacent leg/hypotenuse. Multiply 10 by 4/5 to get 8. The hypotenuse is not counted as the adjacent leg in this ratio.

The hypotenuse also touches the acute angle but is not the adjacent leg in this definition. Identify the two legs first and keep the right-triangle condition explicit.

  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.
Worked example

See the method in action

Teaching case: For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 10 cm. Find the adjacent leg. Candidate answers: (A) 16 cm (B) 7 cm (C) 8 cm (D) 9 cm. Solution: Cosine is adjacent leg/hypotenuse. Multiply 10 by 4/5 to get 8. The hypotenuse is not counted as the adjacent leg in this ratio. The final answer is 8 cm. The hypotenuse also touches the acute angle but is not the adjacent leg in this definition. Identify the two legs first and keep the right-triangle condition explicit.

Your turn

Check your reasoning

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Helpful tips
  • 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.
For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 15 cm. Find the adjacent leg.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

12 cm

Cosine is adjacent leg/hypotenuse. Multiply 15 by 4/5 to get 12. The hypotenuse is not counted as the adjacent leg in this ratio.

Take it further

Changed-input task: 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. Show the mathematical reason, not only the selected answer. Expected result: 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.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 20 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.
2. For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 25 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.
3. For an acute angle θ in a right triangle, cos θ=4/5 and the hypotenuse is 30 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.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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