Find the area of the quadrilateral shown.

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.

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.

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

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