Question

Let A and B be two regular polygons having a and b sides, respectively. If b = 2a and each interior angle of B is \(\frac{3}{2}\) times each interior angle of A, then each interior angle, in degrees, of a regular polygon with a + b sides is

• A. 120
• B. 150
• C. 160
• D. 130

Answer 1

The Correct Option is B

Step 1: Interior Angle of Polygon A 

Polygon A has \( a \) sides.
Interior angle formula: \[ \theta_A = \left(\frac{(a - 2) \times 180}{a}\right)^\circ \]

Step 2: Interior Angle of Polygon B

Given: \( b = 2a \).
Interior angle of polygon B: \[ \theta_B = \left(\frac{(b - 2) \times 180}{b}\right)^\circ = \left(\frac{(2a - 2) \times 180}{2a}\right)^\circ \]

Step 3: Relationship Between Angles

Given: \( \theta_B = \frac{3}{2} \times \theta_A \) \[ \frac{(2a - 2) \times 180}{2a} = \frac{3}{2} \times \frac{(a - 2) \times 180}{a} \] Cancel 180 from both sides: \[ \frac{2a - 2}{2a} = \frac{3}{2} \times \frac{a - 2}{a} \] Cross-multiply: \[ 2(2a - 2) \cdot a = 3(2a)(a - 2) \] Simplify: \[ 4a^2 - 4a = 6a^2 - 12a \Rightarrow 2a^2 - 8a = 0 \Rightarrow 2a(a - 4) = 0 \] So, \( a = 0 \) or \( a = 4 \). Since polygons can't have 0 sides, we use \( a = 4 \).

Step 4: Find Interior Angle of Polygon with \( a + b = 4 + 8 = 12 \) Sides

\[ \theta_C = \left(\frac{(12 - 2) \times 180}{12}\right)^\circ = \left(\frac{10 \times 180}{12}\right)^\circ = 150^\circ \]

Final Answer:

Each interior angle of the polygon with \( a + b = 12 \) sides is 150 degrees.