Question

select the correct answer from each drop-down menu. given: △abc with altitude h shown in the diagram. the altitude is perpendicular to \\(\overline{bc}\\). prove: area of \\(\triangle abc = \frac{1}{2}ab\sin(c)\\) by the definition of the \\(\boldsymbol{\text{drop-down}}\\), the area of \\(\triangle abc\\) is \\(\frac{1}{2}ah\\). then, \\(\sin(c) = \frac{h}{b}\\) by the definition of the sine ratio. using the \\(\boldsymbol{\text{drop-down}}\\), \\(b\sin(c) = h\\). lastly \\(a = \frac{1}{2}ab\sin(c)\\) by the \\(\boldsymbol{\text{drop-down}}\\).

Explanation:

Step1: Identify area formula

Area of triangle is $\frac{1}{2} \times \text{base} \times \text{height}$ (definition of triangle area).

Step2: Relate h and sin(C)

$\sin(C) = \frac{h}{b}$ (sine ratio in right triangle).

Step3: Solve for h

Multiply both sides by b: $h = b\sin(C)$ (algebraic manipulation).

Step4: Substitute h into area

Area = $\frac{1}{2}ab\sin(C)$ (substitution property).

Answer:

First drop-down: area of a triangle
Second drop-down: multiplication property of equality
Third drop-down: substitution property of equality