# Determining tangent lines: lengths

Solve two problems that apply properties of tangents to determine if a line is tangent to a circle.

## Problem 1

Segment start overline, O, C, end overline is a radius of circle O.
Note: Figure not necessarily drawn to scale.
Is line tangent to circle O?

A start color red, l, i, n, e, end color red that is tangent to a circle at a particular point is perpendicular to the start color gray, r, a, d, i, u, s, end color gray at that point.
Let's check to see whether is tangent by using the converse pythagorean theorem to determine if triangle, A, O, C is a right triangle.
We will need to know the length of start overline, A, O, end overline.
start color purple, B, O, equals, 5, end color purple because it is a radius just like start color pink, C, O, end color pink, so:
A, O, equals, start color blue, 8, end color blue, plus, start color purple, 5, end color purple, equals, 13
Now we can use the Pythagorean theorem to determine whether triangle, A, O, C forms a right triangle.
\begin{aligned} \green{AC}^2 + \pink{OC}^2 &\stackrel{?}{=} AO^2\\ \green{11}^2+ \pink{5}^2 &\stackrel{?}{=} 13^2 \\ 146 &\neq 169 \end{aligned}
So, triangle, A, O, C is not a right triangle.
No, is not tangent to circle O, because start overline, A, C, end overline is not perpendicular to start overline, O, C, end overline.

## Problem 2

Segment start overline, O, C, end overline is a radius of circle O.
Note: Figure not necessarily drawn to scale.
Is line tangent to circle O?
\begin{aligned} \green{AC}^2 + \pink{OC}^2 &\stackrel{?}{=} AO^2\\ \green{16}^2+ \pink{12}^2 &\stackrel{?}{=} 20^2 \\ 400 &= 400 \end{aligned}