Use Opposite over Adjacent Leg

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: For an acute angle θ, tan θ=3/4 and its adjacent leg is 8 cm. Find its opposite leg. Candidate answers: (A) 12 cm (B) 5 cm (C) 6 cm (D) 7 cm.
• Independent check inputs: For an acute angle θ, tan θ=3/4 and its adjacent leg is 12 cm. Find its opposite leg. Candidate answers: (A) 10 cm (B) 18 cm (C) 8 cm (D) 9 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. Tangent is the ratio of the opposite leg to the adjacent leg for an acute angle. It links the two legs directly, so a hypotenuse measurement is not needed.

Here is the method on the supplied teaching case. Tangent is opposite/adjacent. Solve o/8=3/4, giving o=8×3/4=6 cm. No hypotenuse is used in this ratio.

Keep the reference angle fixed while identifying the legs. A tangent ratio is dimensionless, but the recovered length keeps the input length unit; using cosine or sine would involve a different denominator.

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: 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. Show the mathematical reason, not only the selected answer. Expected result: 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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