Question

(a) Can Kayla conclude that and are similar? Why or why not? HELP PLEASE

1. Kayla wants to find the width, AB, of a river. She walks along the edge of the river 100 ft and marks point C. Then she walks 22 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.

(a) Can Kayla conclude that and are similar? Why or why not?
(b) Suppose DE = 32 ft. What can Kayla conclude about the width of the river? Explain.
Answer:

Answer 1

See below:

Explanation:

Let's solve `a)` and `b)` separately.

`color(red)"a)"` How do we know if the triangles are equal? Well, all they need to be similar are 2 equal angles (also called AA Similarity Theorem).

`angle B = angle D` because they are both right angles
`angle ACB = angleDCE` because they are vertical angles.

So, we can conclude that the triangles are similar by AA Similarity Theorem.

`color(red)"b)"` Since the triangles are similar, they're also proportional. We can compare the lengths of the corresponding sides so that they are equal in proportion:

`(DE)/(AB) = (DC)/(CB)`

Substitute in the lengths they have given:

`32/(AB) = (22)/(100)`

Cross multiply and solve:

`32/(AB) = (22)/(100)`

`AB xx 22 = 32 xx 100`

`22AB = 3200`

`AB = 145.5`

So the length of the river (`AB`) is `145.5` feet.