On a straight line passing through
the foot of a tower, points 𝐶 and 𝐷 are at distances of four metres and 16 metres
from the foot, respectively. If the angles of elevation from 𝐶
and 𝐷 to the top of the tower are complementary, find the height of the tower.
Here’s our tower and our straight
line passing through the foot of the tower. We have point 𝐶 and point 𝐷. Point 𝐶 is four metres from the
foot of the tower and point 𝐷 is 16 metres from the foot of the tower.
Here are the two angles of
elevation. We’re told that these two angles
are complementary. We know they add to 90 degrees. We’ll call angle 𝐶 𝜃 and that
makes angle 𝐷 90 minus 𝜃. Our unknown value is ℎ, the height
of this tower. We’re dealing with two right
triangles. And we have opposite side measures
and adjacent side measures.
Opposite and adjacent side lengths
tell us that we’ll need to be working with a tangent ratio. Let’s set up a tangent ratio for
the smaller of the two triangles. Tangent of 𝜃 here is equal to the
height of the tower over four. We’ll set up the same ratio for our
larger triangle. This time we have tangent of 90
minus 𝜃 equals ℎ the height of the tower over 16.
And here is where we need to know
some properties of tangent. The tangent of angle 𝜃 is equal to
the reciprocal of tangent of 90 minus 𝜃. What is the reciprocal of tangent
of 90 degrees? That would be the cotangent of 90
minus 𝜃. Cotangent of 90 minus 𝜃 equals the
adjacent side length over the opposite side length. It’s the reciprocal of tangent. In our case, it would be 16 over
Based on that property, these two
values are equal to each other. ℎ over four must be equal to 16
over ℎ. We can use this equality to solve
for ℎ. First, we get the ℎ out of the
denominator by multiplying both sides of the equation by ℎ. On the left, ℎ squared over four
equals? On the right, the ℎs cancel out,
leaving us with 16. From there, we multiply by four on
both sides. And we find that ℎ squared equals
16 times four, 64. We take the square root of ℎ
squared and the square root of 64. And the height would be equal to
plus or minus eight.
But because we know we’re measuring
the height and negative ℎ value doesn’t make sense, our ℎ then must be positive
eight. And in context of our question, it
means that the tower is eight metres tall.