Question

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Emma wants to measure the distance across a river between two large rocks, point A and point B. From B, facing A she turns 90° and walks 79 feet to point C where there is a post. She then continues straight 23 feet to a tree stump at point D. Then, turning 90° again she walks 57 feet to point E so that she is on a straight line between the post C and the rock A on the other side of the river as shown in the diagram. Answer each of the following questions. Show all your work. Five points each. 15 points total. Triangle ABC and EDC are similar triangles. What triangle similarity theorem or postulate will prove this? Why? State which sides and angles you would use to make the argument. Since the triangles are similar the corresponding sides are proportional. Write a proportion you could use to find the unknown distance from A to B using the three known distances. Use “cross multiply and divide” to solve this proportion for the unknown, and state the distance from A to B to the nearest whole foot.

Answer 1

I made a diagram from your description.
Notice that booth ∠ACB and ∠DCE are vertical angles, and we know that vertical angles are congruent by the vertical angles theorem. Also, since she turned around 90° from B towars D and from D towards E, ∠ABC and ∠CDE are right angles, and we also know that right triangles are congruent.
So far we prove that ∠ACB≅∠DCE and ∠ABC≅∠CDE, and since both angles are corresponding congruent angles, we just prove that △ABC and △EDC are similar by the AA postulate.

The corresponding sides we are interested in are AB, BC, ED, and DC:

`(AB)/(BC) = (ED)/(DC)`

`(AB)/(79) = (57)/(23)`

Now the only thing is cross multiply and divide to find the length of AB:

`AB= ((57)(79))/(23) =195.87`
We can conclude that the distance from A to B to the nearest whole foot is 196 feet.