Question

PART E.)In terms of the trigonometry ratios for triangle BCD, what is the length of line BD. Insert text on the triangle to show the length of line BD. When you’re done use the formula for the area of a triangle area equals 1/2 times base times height write an expression for the area of triangle ABC this when you do this base your answer on what u did in part E

Answer 1

Sine formula

`\sin (angle)=\frac{\text{opposite side}}{hypotenuse}`

Considering angle C from triangle BCD, the opposite side is side BD and the hypotenuse is side BC which length is a units. Then:

`\begin{gathered} \sin (\angle C)=(BD)/(a) \\ \text{ Isolating BD} \\ \sin (\angle C)\cdot a=BD \end{gathered}`

The area of a triangle is calculated as follows:

`A=(1)/(2)\cdot\text{base}\cdot\text{height}`

In triangle ABC the base is b units long and its height is segment BD, then the area of triangle ABC is:

`\begin{gathered} A=(1)/(2)\cdot b\cdot BD \\ \text{ Substituting with the previous result:} \\ A=(1)/(2)\cdot b\cdot a\cdot\sin (\angle C) \end{gathered}`