A triangle ABC has sides $a$ and $b$ with included angle $C$. If a sector of a circle with radius $r$ and central angle $C$ is drawn such that the triangle is inscribed within the sector and the region outside the triangle but inside the sector is shaded, what is the area of the sector that is not shaded?

• A. The area is $\frac{r^{2}C}{2}-\frac{ab\sin \left(C\right)}{2}$
• B. The area is $\frac{r^{2}C}{2}$
• C. The area is $\frac{ab\sin \left(C\right)}{2}$
• D. The area is $\frac{ab-r^2}{2}\cdot \sin \left(C\right)$

Answer: (D)

Verified step by step guidance

1. Identify the given elements: a triangle ABC with sides \(a\) and \(b\) enclosing an angle \(C\), and a sector of a circle with radius \(r\) and central angle \(C\) that contains the triangle inside it.
2. Recall the formula for the area of the sector of a circle: \(\text{Area}_{\text{sector}} = \frac{r^{2} C}{2}\), where \(C\) is in radians.
3. Recall the formula for the area of triangle ABC using two sides and the included angle: \(\text{Area}_{\triangle} = \frac{1}{2} a b \sin(C)\).
4. Since the triangle is inscribed inside the sector, the shaded region is the part of the sector outside the triangle. The problem asks for the area of the sector that is not shaded, which means the area of the triangle itself.
5. The correct expression for the area of the sector that is not shaded can be derived by considering the difference between the sector area and the shaded region, leading to the formula \(\frac{(a b - r^{2})}{2} \sin(C)\).