First slide

Arithmetic progression

Question

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

Moderate

A

$\frac{3}{5}, \frac{4}{5}$
B

$\frac{1}{\sqrt{3}}, \sqrt{\frac{2}{3}}$
C

$\frac{1}{2}, \frac{\sqrt{3}}{2}$
D

none of these

Solution

Let $∠ C = 90^{\circ}$ being greatest and $B = 90^{\circ} - A$ .
The sides are a - d, a, and a + d
We have $(a + d)^{2} = (a - d)^{2} + a^{2}$ (using Pythagoras theorem)
$∴ 4 ad - a^{2} = 0 or a = 4 d$
Hence, the sides are 3d, 4d, 5d
Clearly,
$\begin{matrix} \sin A = \frac{BC}{AB} = \frac{a - d}{a + d} = \frac{3 d}{5 d} = \frac{3}{5} \\ \sin B = \frac{AC}{AB} = \frac{a}{a + d} = \frac{4 d}{5 d} \\ = \frac{4}{5} \end{matrix}$

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