Two Tangent Lines to a Circle Intersect at an External Point

In this video you will learn about the relationships that exist when two tangent lines to a circle intersect at an external point.

In this video, we are going to look at what happens when two tangent lines intersect at a common external point. Here we have circle $O$ and point $P$ . Let’s draw two tangent lines from point $P$ to circle $O$ . When this happens, the two tangents are congruent to each other.

Let’s name the two points where the tangent intersects the circle as A and B. So:
$\overline{AP}\cong\overline{BP}$

If we draw a radius to points $A$ and $B$ , then we know that a radius and a tangent line are perpendicular to each other, giving us right angles. Now, we are going to draw another line from point $P$ to the center of circle $O$ . This gives us two triangles that are congruent because we have a right angle that is congruent in both triangles, a tangent line that is congruent in both triangles, a radius that is congruent in both triangles, and a line that is shared so it is also congruent in both triangles. Since we know that the two triangles are congruent, we also know that:
$\angle{APO}\cong\angle{BPO}$
$\angle{AOP}\cong\angle{BOP}$
When drawing two tangent lines to a circle that intersect at a common point, we end up with two congruent right triangles. This means that if we need to find a missing side, we can use the Pythagorean Theorem.