First slide

Arithmetic progression

Question

If the sides of a triangle are in AP, and the greatest angle of the triangle is double the smallest angle, the ratio of the sides of the triangle is

Moderate

A

3:4:5
B

4:5:6
C

5:6:7
D

7:8:9

Solution

Let the sides of the triangle be a-d, a and a+d, with a > d>0.
Clearly, a -d is the smallest side and a + d is the largest side.
So, A is the smallest angle and C is the largest angle. It is given that C = 2A.
Thus, the angles of the triangle are A, 2A and $π$ -3A.
Applying the law of sines, we obtain
$\begin{aligned} \frac{a - d}{\sin A} = \frac{a}{\sin (π - 3 A)} = \frac{a + d}{\sin 2 A} \\ \Rightarrow & \frac{a - d}{\sin A} = \frac{a}{\sin 3 A} = \frac{a + d}{\sin 2 A} \\ \Rightarrow & \frac{a - d}{\sin A} = \frac{a}{3 \sin A - 4 \sin^{3} A} = \frac{a + d}{2 \sin Acos A} \\ \Rightarrow & \frac{a - d}{1} = \frac{a}{3 - 4 \sin^{2} A} = \frac{a + d}{2 \cos A} \\ \Rightarrow & 3 - 4 \sin^{2} A = \frac{a}{a - d} and 2 \cos A = \frac{a + d}{a - d} \\ \Rightarrow & 4 \cos^{2} A - 1 = \frac{a}{a - d} and 2 \cos A = \frac{a + d}{a - d} \end{aligned}$
$\Rightarrow {(\frac{a + d}{a - d})}^{2} - 1 = \frac{a}{a - d} \Rightarrow a = 5 d$
Thus, the sides of the triangle are a-d, a, a + d i.e. 4d, 5d, 6d.
Hence, the ratio of the sides of the triangle is 4d :5d: 6d i.e. 4:5:6.

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App