Chapter 11, Problem 3STP

To determine

To find: The area of the given triangle.

Answer to Problem 3STP

The area of the triangle is option $\left(\text{D}\right)171.9{\text{ in}}^{\text{2}}$ .

Explanation of Solution

Given information:

The figure of a triangle with dimensions is given as follows:

  

Figure (1)

Formula used:

(i) Pythagorean theorem for right angled triangle:

  ${\text{Hypontenuse}}^2={\text{Adjacent}}^2+{\text{opposite}}^2$ .

(ii) The tangent formula for a right angled triangle:

  $\tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}$

(iii) Area of the triangle is $A=\frac{1}{2}bh$ ; where b is the base and h is the height.

Calculation:

Use the Pythagoras theorem to compute the value of y.

  $\begin{array}{c}{17}^2=y^2+{13}^2\\ 289=y^2+169\\ y=\sqrt{289-169}\\ =10.954\end{array}$

Calculate the value of x using the tangent formula as follows:

  $\begin{array}{c}\tan 40°=\frac{\text{13}}{x}\\ x=\frac{\text{13}}{\tan 40°}\\ =\frac{\text{13}}{0.8391}\\ \simeq 15.493\end{array}$

The area of the required triangle is computed as follows:

  $\begin{array}{c}A=\frac{1}{2}\left(10.954+15.493\right)\left(13\right)\\ =\frac{1}{2}\left(26.447\right)\left(13\right)\\ =\frac{1}{2}\left(343.811\right)\\ =171.9055\end{array}$

Round off the area value to the nearest tenth, then $\text{area}=171.9{\text{ in}}^{\text{2}}$ .

Therefore, the area of the triangle is $171.9{\text{ in}}^{\text{2}}$ .