Problem: Pythagorean Theorem and Tangent-Secant

There are multiple ways to solve problems with circles, but some will be more straightforward than others, based on the information given.

If a circle has a tangent drawn to it, then that tangent is perpendicular to the radius it intersects. In other words, a right angle is created. Generally speaking, that right angle will probably signal the need to use the Pythagorean Theorem at some point in the problem.

On the other hand, if the tangent is accompanied by a secant line, then a second theorem can be plucked from our toolkit: the Tangent-Secant Theorem. If the tangent and secant intersect at a point outside the circle, then the square of the length of the tangent from the external point to the circle will be equal to the product of the portion of the secant outside the circle times the length of the entire segment.

Consider the problem below:

Which of the two theorems do we need to use?

The answer is: either one of them.

The circle has three radii drawn, but only one is labeled. Write the "6" next to the other two segments.

You can now solve for x using the right triangle with legs 6 and x, and with hypotenuse 10. Or you can solve for tangent with length x using the secant with a length of 16 and an external length of 4.

If you choose to work both of them out, you'll find the same answer.

Keeping that in mind, look at this next problem:

Now you will see that you can make a diameter from the given radius, and create a secant. This gives you a second option for some for x.