Question

which triangles area can be calculated using the trigonometric area formula?

Explanation:

Step1: Recall trigonometric area rule

The trigonometric area formula for a triangle is $Area=\frac{1}{2}ab\sin C$, where $a$ and $b$ are two sides of the triangle, and $C$ is the included angle between them. We need to identify which triangle has two known sides and their included angle.

Step2: Analyze each triangle

1. Triangle ABC: Only all three sides are given (SSS), no included angle provided.
2. Triangle KLM: Two sides ($KL=29$, $KM=15$) and the included angle $\angle K=35^\circ$ are given. This matches the requirements for the trigonometric area formula.
3. Triangle DEF: One side ($EF=20$) and two non-included angles are given, no pair of sides with included angle.
4. Triangle GHJ: Two sides ($GH=17$, $GJ=22$) and a non-included angle $\angle J=47^\circ$ are given, not the included angle between the two sides.

Answer:

The triangle KLM (the second triangle, with sides 29, 15 and included 35° angle)