Question

consider the diagram and the derivation below. given: in △abc, ad ⊥ bc. derive a formula for the area of △abc using angle c. it is given that in △abc, ad ⊥ bc. using the definition of sine with angle c in △acd results in sin(c) = h/b. using the multiplication property of equality to isolate h, the equation becomes bsin(c) = h. knowing that the formula for the area of a triangle is a = 1/2bh is and using the side lengths as shown in the diagram, which expression represents the area of △abc? 1/2bsin(c) 1/2absin(c) 1/2cbsin(c) 1/2hbsin(c)

Explanation:

Step1: Recall area formula

The area formula of a triangle is $A=\frac{1}{2}bh$.

Step2: Substitute $h$

We know that $h = b\sin(C)$. Substituting $h$ into the area - formula, we get $A=\frac{1}{2}b\times b\sin(C)=\frac{1}{2}ab\sin(C)$ (where the base is $a$ and the height $h$ related to angle $C$ is $b\sin(C)$).

Answer:

$\frac{1}{2}ab\sin(C)$