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.
