Multiple choice

If the sides of a right -angled triangle are in A.P., then the sines of the acute angles are

  1. $\cfrac 3 5, \cfrac 4 5$
  2. $\cfrac {1 } {\sqrt 3}, \sqrt { \cfrac { 2 }{ 3 } } $
  3. $\cfrac { 1 }{ 2 } ,\cfrac { \sqrt { 3 } }{ 2 } $

  4. None of these
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A Correct answer
Explanation

If sides are in AP, they can be represented as a-d, a, a+d. For a right triangle, (a-d)^2 + a^2 = (a+d)^2. a^2 - 2ad + d^2 + a^2 = a^2 + 2ad + d^2. a^2 = 4ad => a = 4d. Sides are 3d, 4d, 5d. The sines of the acute angles are 3/5 and 4/5.

AI explanation

Let the sides of the right-angled triangle be a-d, a, and a+d, where a+d is the hypotenuse. Using the Pythagorean theorem, we write (a-d)^2 + a^2 = (a+d)^2, which expands and simplifies to a = 4d. Substituting this back gives side lengths of 3d, 4d, and 5d, making the sides a 3-4-5 triangle. The sines of the acute angles are therefore 3/5 and 4/5.

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