Exercises: Derive the Formula A = (1/2)ab sin(C)

In the derivation of $A = \dfrac{1}{2}ab\sin(C)$, an altitude $h$ is drawn from vertex $B$ perpendicular to side $b$ (= $AC$). In the right triangle formed at vertex $C$, the sine definition gives $\sin(C) = \dfrac{h}{a}$. Solving for $h$: $h =$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . Substituting into $A = \dfrac{1}{2} \cdot b \cdot h$ gives $A =$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Find the area of a triangle with sides $a = 5$ cm and $b = 7$ cm and included angle $C = 60\degree$. Use $\sin(60\degree) \approx 0.866$. Round to the nearest tenth.

Triangle $PQR$ has sides $PQ = 10$ and $PR = 12$, with included angle $P = 50\degree$. Find the area of triangle $PQR$. Use $\sin(50\degree) \approx 0.766$. Round to the nearest tenth.

A triangle has sides $a = 6$ m and $b = 8$ m with included angle $C = 120\degree$. Find its area. Use $\sin(120\degree) \approx 0.866$. Round to the nearest tenth.

Find the area of a triangle with two sides of length 4 ft and 9 ft and an included angle of 45°. Use $\sin(45\degree) \approx 0.707$. Round to the nearest tenth.