Question

A summer camp counselor wants to find a length, $x$, in feet, across a lake as represented in the sketch above. The lengths represented by $AB$, $EB$, $BD$, and $CD$ on the sketch were determined to be $1800$ feet, $1400$ feet, $700$ feet, and $800$ feet, respectively. Segments $AC$ and $DE$ intersect at $B$, and $\angle AEB$ and $\angle CDB$ have the same measure. What is the value of $x$?

• A. $1600$
• B. $1500$
• C. $1400$
• D. $1300$

Answer 1

The correct option is A $1600$

If angles $E$ and $D$ are the same with the lines intersecting at $B$, then that means that $A$ and $C$ are also the same, as the opposite angles formed at point $B$ are also the same, so you have two similar triangles.
Similar triangles have proportional lengths. So we know length $CD$ is $800$ ft and its similar leg on the other side ($AE$) is unknown.
So we need to find two known lengths, one on the large and one on the small, that are similar.
$BE$ is $1400$ ft and $BD$ is $700$ ft. Those are you two known similar legs. So now we can use them in a proportion to find $AE$.
We will do the line ending at $B$ over the line that doesn't include $B$.
Large triangle: $\frac{1400}{x}$

Small triangle: $\frac{700}{800}$
Set them up in a proportion:
$\frac{1400}{x}$$=\frac{700}{800}$

Solve for $x$:
$\Rightarrow 7x=11200$
$\Rightarrow x=1600$ ft