Answered

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A telegraph pole stands on a line joining two points P and Q on the ground.
The telegraph pole is 9 m tall. The angle of depression of the top of the pole to point P
is 4°. The angle of depression of the top of the pole to point Q is 7º. What is the
horizontal distance between P and Q.


Sagot :

reinhard10158

Step-by-step explanation:

let's call the top of the pole T, and the ground point of the pole G.

so, the situation gives us 2 right-angled triangles:

PTG and QTB.

from both triangles we know one leg (TG, the pole = 9 m), the inner angles at G (90° in both cases), and actually the inner angles at T, because we know the angles of depression there.

the inner triangle angles at T are just the complementary angles of the angles of depression (that means they add up together to 90°).

the inner angle at PTG = 90 - 4 = 86°

the inner angle at QTG = 90 - 7 = 83°

and because the sum of all angles in a triangle is always 180°, we even know the third angles at P and Q :

angle P = 180 - 90 - 86 = 4°

angle Q = 180 - 90 - 83 = 7°

yes, they are the same as the angles of depression (must be the same).

so, we know from both triangles 1 side and all 3 angles.

with the law of sine we can get every missing side.

particularly we need the second legs (sides in the ground) to get the distance between both points.

a/sin(A) = b/sin(B) = c/sin(C)

a, b, c being the sides always opposite of their related angles.

TG/sin(4) = PG/sin(86)

PG = 9×sin(86)/sin(4) = 128.7059963... m

TG/sin(7) = QG/sin(83)

QG = 9×sin(83)/sin(7) = 73.29911785... m

the distance between P and Q can have now 2 solutions :

if P and Q are on different sides of the pole, then

PQ = 128.7059963... + 73.29911785... = 202.0051142... m

but if they are on the same side of the pole, then their distance is

PQ = 128.7059963... - 73.29911785... = 55.40687846... m

due to the phrasing of the first sentence I suspect the first solution to be right.

but just in case, I gave you also the second possible solution.

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