# Video: CBSE Class X • Pack 2 • 2017 • Question 13

CBSE Class X • Pack 2 • 2017 • Question 13

03:39

### Video Transcript

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 ℎ.

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.