Using Properties of Tangents to Solve for Unknown Lengths

Daniel needs to figure out the amount of grain that is held in a silo. Before he can do so, he must know the radius of the silo. He currently knows that when he is standing approximately 8 feet away from the silo he is approximately 16 feet away from a point of tangency of the silo.

Before we investigate this problem further, let's look at a few example problems that might aid us in our investigation.

Given the diagrams below, find the unknown lengths. Show all work.

We know that a tangent line is perpendicular to the radius of a circle, therefore we can just use the Pythagorean Theorem to solve for the unknown length. So we have

a² = 6² + 8²

a² = 36 + 64

a² = 100

a = 10

 

Again, we know that a tangent line is perpendicular to the radius of a circle, therefore we can just use the Pythagorean Theorem to solve for the unknown length. Now, we are only looking for the length x, outside the circle. So let r be the radius of the circle, and let y = x = r. So we have

y² = 12² + 16²

y² = 144 + 256

y² = 400

y = 20

Therefore, since y = x + r and we already know that r = 12, we have that

x + 12 = 20

x = 20 - 12

x = 8

Now, we know that any two tangent lines of a circle are equal in length. So we must have that

2x + 10 = 3x + 7

2x + 3 = 3x

3 = x

Now try drawing a diagram for the given problem.

Now, how might you use what you know about tangents of circles to go about solving this problem?

We know that tangents are perpendicular to the radius of a circle, wo we know we can use the pythagorean theorem to find the length of the radius. We would have one side as r, one side as 16 ft, and the hypotenuse would be (r + 8) ft.

What is the length of the radius of the silo? Show your work.

(r + 8)² = r² + 16²

r² + 16r + 64 = r² + 256

16r = 192

r = 12

Therefore, the radius of the silo is 12 ft.

 

 

Developed by Katherine Huffman