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 $21$.