Question

Find the area of the quadrilateral shown.

Short answer

The area of the quadrilateral is $54.15$square units.

Step-by-step solution

Step 1. Given Information

The given sides of the quadrilateral are $5,7,8,\mathrm{and}11.$The angles of the quadrilateral are$72°\mathrm{and}90°.$We have to find the area of the quadrilateral.

Step 2. Finding the area

To find the area, we will divide the quadrilateral into two triangles.

So, the first triangle with sides $5,11,$and angle $72°.$

Now, to find the area of the triangle we will use the formula of the product of the sides and sine of the included angle.

$$\begin{gathered} A_1=\frac{1}{2}\left(5\right)\left(11\right)\sin 72° \\ {A}_1=\frac{55}{2}\sin 72° \\ {A}_1=27.5\left(0.95\right) \\ {A}_1\approx 26.15 \end{gathered}$$

Step 3. Finding the area

Now, the second triangle is with sides $7,8,$and angle $90°.$

Use the formula of the product of sides and sine of the included angle to find the area of the triangle.

$$\begin{gathered} A_2=\frac{1}{2}\left(7\right)\left(8\right)\sin 90° \\ {A}_2=\frac{56}{2} \\ {A}_2=28 \end{gathered}$$

Thus, the area of the quadrilateral is

role="math" localid="1647602092155" $$\begin{gathered} A=A_1+A_2 \\ A=26.15+28 \\ A=54.15\mathrm{square}\mathrm{units} \end{gathered}$$