Question

Kayla wants to find the width, AB, of a river. She walks along the edge of the river 90 ft and marks point C. Then she walks 72 ft further and marks point D. She turns 90° and walks until her location, point A, and point C are collinear. She marks point E at this location, as shown.
Can Kayla conclude that ∆ABC and ∆EDC are similar? Why or why not? Suppose DE = 54 ft. What is the width of the river? Round to the nearest foot. Provide all work.

Answer 1

Part a: The triangles ∆ABC and ∆EDC are similar by AA similarity rule.

Part b: The width of AB is 67.5 feet

Explanation:

Part a: We need to prove that the two triangles ABC and EDC are similar.

To prove the triangles are similar, then their angles must be similar.

Thus, we have,

∠DCE  and  ∠BCA are similar (vertical angles)

∠CDE  and  ∠CBA are similar (right angles)

∠B and ∠A are similar

Hence, the triangles ∆ABC and ∆EDC are similar by AA similarity rule.

Part b: We need to determine the width of AB

Since, the triangles are similar, then their corresponding lengths are proportional.

Thus, we have,

$\frac{DE}{AB} =\frac{DC}{CB}$

where $DE=54 ft$, $DC=72 ft$ and $CB=90ft$

Substituting these values, we get,

$\frac{54}{AB} =\frac{72}{90}$

Multiplying both sides by 90, we get,

$\frac{4860}{AB}=72$

$\frac{4860}{72}=AB$

Dividing, we have,

$67.5=AB$

Thus, the width of the river AB = 67.5 feet