## Area of a Trapezoid

Find the area of a trapezoid where the parallel sides are 16 and 44, and the non-parallel sides are 17 and 25.
Through the vertex of the trapezoid, draw a line parallel to the lateral side.

## Solution

Through vertex $C$ of trapezoid $A B C D (B C=16, A D=44, A B=17, C D=25)$, draw a line parallel to side $A B$ until it intersects the base $A D$ at point $K$.

In triangle $C K D$

$$
C K=17, C D=25, K D=A D-B C=28 .
$$

By Heron's formula

$$
S_{\triangle \mathrm{CKD}}=\sqrt{35 \cdot 7 \cdot 10 \cdot 18}=5 \cdot 7 \cdot 6=210
$$

If $C M$ is the height of this triangle, then

$$
C M=\frac{2 S_{\triangle C K D}}{K D}=\frac{2 \cdot 210}{28}=15
$$

Therefore,

$$
S_{\mathrm{ABCD}}=\frac{A D+B C}{2} \cdot C M=450
$$

Other methods for finding the height $C M$ of triangle $C K D$.

a) Let $D M=x$ and find $x$ from the equation

$$
C D^{2}-D M^{2}=C K^{2}-K M^{2} \text {, or } 25^{2}-x^{2}=17^{2}-(28-x)^{2} .
$$

Then find $C M$ using the Pythagorean theorem in triangle CMD.

b) Find the cosine of angle $C D K$ using the cosine rule in triangle $C D K$. Then compute its sine and find $C M$ from the right triangle CMD.

![](https://cdn.mathpix.com/cropped/2024_05_06_cb0f0d605674d837f9abg-44.jpg?height=352&width=684&top_left_y=990&top_left_x=687)

## Answer

450.