Question

beaver boy has to find the length, in feet, across a lake sketched above to rescue a family on the other side. he measured lengths represented by $overline{ab}$, $overline{eb}$, $overline{bd}$, and $overline{cd}$ and determined them to be 2100 feet, 800 feet, 400 feet, and 300 feet, respectively. segments $overline{ac}$ and $overline{de}$ intersect at b, and angles $angle aeb$ and $angle cdb$ have the same measure. what is the length of $overline{ae}$? (a) 1200 feet (b) 1000 feet (c) 800 feet (d) 600 feet

Explanation:

Step1: Identify similar triangles

Since $\angle AEB=\angle CDB$ and $\angle ABE = \angle CBD$ (vertically - opposite angles), $\triangle AEB\sim\triangle CDB$ by the AA (angle - angle) similarity criterion.

Step2: Set up the proportion

For similar triangles $\triangle AEB$ and $\triangle CDB$, the ratios of corresponding sides are equal. That is, $\frac{AE}{CD}=\frac{EB}{BD}$.

Step3: Substitute the given values

We know that $EB = 800$ feet, $BD = 400$ feet, and $CD = 300$ feet. Substituting these values into the proportion $\frac{AE}{300}=\frac{800}{400}$.

Step4: Solve for $AE$

Cross - multiply: $400\times AE=300\times800$. Then $AE=\frac{300\times800}{400}$. Calculate $300\times800 = 240000$ and $\frac{240000}{400}=600$ feet.

Answer:

D. 600 feet