<p>If the sides of a right angled triangle form an AP, the sines of the acute angles are</p>
<p>(a) \(\frac{3}{5}, \frac{4}{5}\)</p>
<p>(b) \(\frac{1}{3}\)</p>
<p>(c) \(\frac{\sqrt{5} - 1}{2}, \frac{\sqrt{5} + 1}{2}\)</p>
<p>(d) \(\frac{3}{2}, \frac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: If the sides of a right-angled triangle form an AP, we can express them as (a-d), a, (a+d) where a-d and a are legs. Using the Pythagorean theorem and the constraint that these form an AP, we can find the ratio of sides, then calculate the sines of acute angles.
<p><strong>Step 1:</strong> Let the three sides of the right-angled triangle in AP be (a-d), a, and (a+d), where a > d > 0.</p><p><strong>Step 2:</strong> Since these form a right-angled triangle, the largest side (a+d) must be the hypotenuse. The two legs are (a-d) and a.</p><p><strong>Step 3:</strong> Apply the Pythagorean theorem: (a-d)² + a² = (a+d)²</p><p><strong>Step 4:</strong> Expand: a² - 2ad + d² + a² = a² + 2ad + d²</p><p><strong>Step 5:</strong> Simplify: 2a² - 2ad + d² = a² + 2ad + d²</p><p><strong>Step 6:</strong> Reduce: a² = 4ad, which gives a = 4d</p><p><strong>Step 7:</strong> The three sides are: (4d-d) = 3d, 4d, and (4d+d) = 5d, i.e., 3d : 4d : 5d or 3 : 4 : 5.</p><p><strong>Step 8:</strong> This is a 3-4-5 right triangle with hypotenuse 5, and legs 3 and 4.</p><p><strong>Step 9:</strong> For the acute angle opposite to side 3: sin θ₁ = 3/5</p><p><strong>Step 10:</strong> For the acute angle opposite to side 4: sin θ₂ = 4/5</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A