Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{PA}$ is $5,$ and $AB$ has a length of 6. Find the radius of the circle.

AUnVerifedTaxPayer Sep 7, 2024

#1**+1 **

I'm not sure if there is an easier way to do this, but one way would involve trigonometric functions. Since AB = 6, we have an isoceles triangle 5-5-6 at Triangle PAB. Extend the perpendicular bisector of AB and you have two 3-4-5 right triangles. Now you can use sin^-1(4/5) to find Angle PAB. Then you can construct a second right triangle with the hypotenuse being the radius of the circle and one leg being 3, which was the line segment of A to the midpoint of AB. Then you can use tan^-1 to find the radius.

I am not sure if this is the easiest way, but this was all I could think of.

mathrokus Sep 7, 2024