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: Identify base and height

Base = BC = a, height = AD = h

Step2: Express h using sine

From sin(C) = h/b, h = b sin(C)

Step3: Substitute into area formula

Area = (1/2)baseheight = (1/2)a(b sin(C)) = (1/2)ab sin(C)

Answer:

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