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Secant-Tangent Product Theorem: Definition & Use

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will review secants and tangents of circles. After this, we will look at the secant-tangent product theorem, and use examples to show how to use this theorem in general and in real world applications.

Secant and Tangent of a Circle

Imagine you are working with a construction crew. A road already exists through a forest that goes over a circular lake. You want to build another road through a forest that connects to this road, but does not go through the lake.


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As it turns out, the road you will be building and the road it will connect to both represent characteristics of a circle that have their own name. The road/bridge that already exists is called a secant of the circular lake, and the road you're going to build is called the tangent of the circular lake.

In general, a secant of a circle is a line that passes through any two points on the edge the circle, and a tangent of a circle is a line that just touches one point on the edge of the circle. Notice how the road that already exists intersects the circular lake at two points along its shoreline (the start and end of the bridge portion of the road), and the road you will be building just touches the circular lake at one point.

Based on this, we call the road that already exists a secant segment of the circular lake, and we call the road you will be building a tangent segment of the circular lake. Furthermore, we call the portion of the road that already exists that is outside of the lake the external secant segment.


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Secant Tangent Product Theorem

Now for your conundrum: you need to know how long the road you're constructing will be in order to know how many supplies you will need. However, you can't measure it, because it is through a forest, so there are trees and such in the way.

You are able to measure the road that already exists, and you find that the bridge portion of the road is 5 kilometers, and the portion of the road from the bridge to where the new road will be is 4 kilometers.


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Oh boy, how are we going to find the distance of this road? Never fear! Math is here! Thankfully, in mathematics, we have a theorem called the Secant-tangent product theorem, which states that for any secant segment and tangent segment of a circle that meet at a common endpoint outside of the circle, it must be the case that

  • (Length of the whole secant segment)(Length of the external secant segment) = (Length of the tangent segment)2

That is, the product of the length of the whole secant segment and the length of the external segment is equal to the length of the tangent segment squared.


sectanprod4


This is awesome! We can use this to find the length of the road you will be building! We have that the length of the bridge is 5 kilometers, and the length of the existing road outside of the lake is 4 kilometers.

Therefore, the length of the whole secant segment is 5 + 4 = 9 kilometers. If we let the length of the road you're going to build be x, then the secant-tangent product theorem gives:

(9)(4) = x2

Simplifying, we get the following:

x2 = 36 Take the square root of both sides.
x = ±6 It's a distance, so discard the negative option.
x = 6

We get that the distance of the road you will be building will be 6 kilometers. Now you have all the information you need to order supplies. Your boss is going to be so happy!

Another Example

Let's consider one more general example just to make sure we really have the hang of this.


sectanprod5


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