Question

If 7 and 8 are the lengths of two sides of a triangle and \( a \) is the length of its smallest side. The angles of the triangle are in AP and \( a \) has two values \( a_1 \) and \( a_2 \) satisfying this condition. If \( a_1<a_2 \), then \( 2a_1 + 3a_2 = \):

• A. 15
• B. 21
• C. 24
• D. 28

Hint:

In triangles with angles in arithmetic progression, use the Law of Sines and properties of angles to calculate side lengths and other unknowns.

Answer 1

The Correct Option is B

We are given that the lengths of the two sides of the triangle are 7 and 8, and the angles of the triangle are in AP. Let the angles be \( A, B, C \).
Since the angles are in AP, we can write: \[ A = B - d, \quad B = B, \quad C = B + d. \] Using the property of a triangle that the sum of its angles is \( 180^\circ \), we have: \[ A + B + C = 180^\circ. \] Substitute the values of \( A \) and \( C \): \[ (B - d) + B + (B + d) = 180^\circ \quad \Rightarrow \quad 3B = 180^\circ \quad \Rightarrow \quad B = 60^\circ. \] Step 1: Using the Law of Sines Using the Law of Sines, we have: \[ \frac{a}{\sin A} = \frac{7}{\sin B} = \frac{8}{\sin C}. \] Since \( B = 60^\circ \), we know that \( \sin 60^\circ = \frac{\sqrt{3}}{2} \). Therefore, we can calculate the values of \( a_1 \) and \( a_2 \). 
Step 2: Calculating \( 2a_1 + 3a_2 \) After calculating the values of \( a_1 \) and \( a_2 \), we find that: \[ 2a_1 + 3a_2 = 21. \] Thus, the correct answer is \( 21 \).