Chapter 6.3, Problem 58E

To determine

To calculate: The length of two equal sides of an isosceles triangle.

Answer to Problem 58E

The length of two equal sides of an isosceles triangle is $9.9\text{ cm}$ .

Explanation of Solution

Given information:

An isosceles triangle has angle included between its two sides as $\frac{5\pi }{6}$ . Also the area of the triangle is $25{\text{ cm}}^2$ .

Formula used:

The area of a triangle is denoted by $A=\frac{1}{2}ab\sin \theta$ where a andb are the length of its sides and $\theta$ is the included angle.

Calculation:

Consider an isosceles triangle has angle included between its two sides as $\frac{5\pi }{6}$ . Also the area of the triangle is $25{\text{ cm}}^2$ .

Let the length of two equal sides of isosceles triangle is x .

Recall that the area of a triangle is denoted by $A=\frac{1}{2}ab\sin \theta$ where a andb are the length of its sides and $\theta$ is the included angle.

Apply it,

  $\begin{array}{c}24=\frac{1}{2}\left(x\right)\left(x\right)\sin \left(\frac{5\pi }{6}\right)\\ 24=\frac{x^2}{2}\sin \left(150°\right)\end{array}$

Now reference angle for $\theta =150°$ is $30°$ . The terminal end of $\theta =150°$ lies in the Quadrant $\text{II}$ and sine trigonometric function is positive in that quadrant.

  $\begin{array}{c}48=x^2\sin \left(30°\right)\\ 48=x^2\left(\frac{1}{2}\right)\\ x^2=98\\ x=9.9\end{array}$

Thus, the length of two equal sides of an isosceles triangle is $9.9\text{ cm}$ .