Two poles, 30 feet and 50 feet tall, are 40 feet apart and perpendicular to the ground. The poles are supported by wires attached from the top of each pole to the bottom of the other, as in the figure. A coupling is placed at C where the tow wires cross.
Find x, the distance from C to the taller pole?
How high above the ground is the coupling?
How far down the wire from the smaller pole is the coupling?
Answers
Answer:
The measurement is 40 feet.
Step-by-step explanation:
Let A be the distance between the ground and the coupling C (perpendicular to the ground and parallel to both poles), and D be the distance between the tallest poll and the point on the ground from C.
Two similar triangles are formed (Angle-Angle (AA) Similarity).
Finding the distance on the ground from under C to the largest pole using proportions on the smaller pair of triangles.
For the larger pair of triangles, 30d = 40a.
50(40-d) = 40a
Take note of 40a, which substitutes on both equations.
30d=50(40-d)
30a = 2000-50d
80d = 2000d = 25 ft Subtraction from 40 yields the other base, which is 15ft.
Substitute to determine the coupling's height.
Use the Triangle Proportionality Theorem to calculate the distance from C to the taller pole.
Let p denote the distance between C and the taller pole.
The measurement is 40 feet.
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Step-by-step explanation:
Let A be the distance from the ground to the coupling C (perpendicular to the ground and parallel to both poles)
Let D be the distance from the tallest poll to the point on the ground from C.
(Angle-Angle (AA) Similarity) two pair of similar triangles are formed.
Using proportions on the smaller pair of triangles to find the distance on the ground from under C to the largest pole.
%2830%2F40%29+=+%28a%2Fd%29
30d = 40a
for the larger pair of triangles
%2850%2F40%29+=+%28a%2F40-d%29
50(40-d) = 40a
Notice 40a, substituting on both equations
30d)=50(40-d)
30a = 2000-50d
80d = 2000
d = 25 feet. If you subtract from 40 you get the other base which is 15ft.
Substitute to find the height of the coupling
30d = 40a
30(25)=40a
a=750/40
a=18.75 feet height.
How far down the wire from the smaller pole is the coupling?
Notice that the smaller pole formed a dilation by a factor 10 from a 3,4,5 triangle. Therefore, the wire from the smaller pole to the base of the larger is 50ft. Then using the Triangle Proportionality Theorem, formulate the following proportion.
Let y be the distance of the segment from the pole to C (Coupling) use the proportion.
y%2F15=50%2F40
y=%28750%2F40%29
y=18.75 feet
For the distance from C to the Taller pole, using Pythagoras
c%5E2=50%5E2%2B40%5E2 the hypotenuse is about 64 feet round to the unit.
To find the distance from C to the taller pole, again use the Triangle Proportionality Theorem..
Let p be the distance from C to the Taller pole.
p%2F25=64%2F40
The distance is 40 feet.