Finding a Lenght Two vertical poles, one 8 ft tall and the other 24 ft tall, have ropes stretched from the top of each to the base of the other (see the figure). How high above the ground is the point where the ropes cross? [Hint: Use similarity]

Dottie Parra 2021-01-19 Answered
Finding a Lenght Two vertical poles, one 8 ft tall and the other 24 ft tall, have ropes stretched from the top of each to the base of the other (see the figure). How high above the ground is the point where the ropes cross? [Hint: Use similarity]
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SabadisO
Answered 2021-01-20 Author has 108 answers

Given:
One pole is 8 ft tall and other pole is 24 ft tall.
Both have ropes stretched from the top of each to the base of the other.
image
Figure(1)
Approach:
If all three angles of a triangle are equal to corresponding angles of the other triangle, then both the triangles are similar and the rule is referred to as AAA rule.
Calculation:
By AAA rule, triangle AEF is similar to triangle ADC.
So,
AEh=AE+EDDC
=AE+ED24...(1)
By AAA Rule, triangle BAD is similar to triangle FED.
So,
EDh=AE+EDAB
=AE+ED8
AE+ED=8EDh...(2)
Substitute value AE+ED from equation (2) in equation (1).
AEh=8EDh/24=8ED24h
AE=ED3
Substitute ED3 for AE in Equation (2).
AE3+ED=8EDh
4ED3=8EDh
43=8h
h=3×84
=6
Therefore, the height of the point above the ground where the ropes cross is 6 ft.

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