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

Step2: Substitute $h$

We know that $h = b\sin(C)$. Substitute $h$ into the area - formula: $A=\frac{1}{2}b\times b\sin(C)$ is incorrect. The correct base - height relationship for $\triangle ABC$ with base $a$ and height $h$ (where $h = b\sin(C)$) gives $A=\frac{1}{2}ab\sin(C)$.

Answer:

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