Given a triangle with sides $a$ and $b$ and included angle $C$, which formula correctly gives the area of the triangle using the SAS (Side-Angle-Side) method?

• A. $\frac{a\times b}{2}$
• B. $\frac{a+b}{2}$
• C. $\frac{1\times a\times b\times \sin \left(C\right)}{2}$
• D. $\frac{a\times b\times \cos \left(C\right)}{2}$

Answer: (C)

Verified step by step guidance

1. Recall that the area of a triangle can be found using the formula involving two sides and the included angle (SAS): \(\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)\).
2. Identify the given elements: sides \(a\) and \(b\), and the included angle \(C\) between them.
3. Understand that the sine function is used here because it relates the height of the triangle to the sides and the included angle, effectively giving the perpendicular height when multiplied by one side.
4. Write down the formula explicitly: \(\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)\).
5. Note that other options involving addition of sides or cosine of the angle do not correctly represent the area in the SAS context.