If an angle of a triangle remains unchanged but each of its two including sides is doubled, then by what factor does the area get multiplied?

### Concept & Triangle Area with Sine Rule The area of a triangle can be found using the lengths of two sides and the sine of the included angle: $$Area = \frac{1}{2} \cdot a \cdot b \cdot \sin(\theta)$$ ### Step-by-Step Solution * Let the original including sides be $a$ and $b$, and the included angle be $\theta$. * Original Area ($A_1$) = $\frac{1}{2} \cdot a \cdot b \cdot \sin(\theta)$. * The new sides are doubled, so they become $2a$ and $2b$. The angle $\theta$ remains unchanged. * New Area ($A_2$) = $\frac{1}{2} \cdot (2a) \cdot (2b) \cdot \sin(\theta)$. * $A_2 = 4 \cdot (\frac{1}{2} \cdot a \cdot b \cdot \sin(\theta))$. * $A_2 = 4 \cdot A_1$. * The area is multiplied by a factor of 4. ### Exam Strategy & Shortcut Area is a 2D measure dependent on the product of two linear dimensions. If both dimensions are scaled by a factor of 2, the area scales by $2 \times 2 = 4$. The angle is irrelevant as long as it remains constant. ### Common Pitfall Thinking the area only doubles because the sides are doubled, failing to recognize that the area is proportional to the *product* of the two sides. ### Final Answer Therefore, the correct answer is **4**.