# Tangent to a Circle

How to solve the tangent to a circle problems: definition, properties, examples, and their solutions.

## Definition

A tangent to a circle is a line

that touches the circle.

From a point exterior of the circle,

you can draw two tangents.

## Property 1

The tangent and the radius are perpendicular

at the intersecting point of the circle.

## Example 1

Ray *AB* is tangent to the circle.

So ray *AB* and the radius are perpendicular.

The radius is 5.

So the green segment is 5.

See this right triangle.

Starting from the shortest side,

the sides of the right triangle are

(5, *AB*, 12).

Then the related Pythagorean triple is

(5, 12, 13).

So *AB* = 12.

Pythagorean triples

## Property 2

If two segments from an exterior point

are tangent to a circle,

then those two segments are congruent.

## Example 2

The blue segments

start from the same point *A*

and are tangent to the same circle.

So the blue segments are congruent.

So the blue segments are both 7.

The green segments

start from the same point *B*

and are tangent to the same circle.

So the green segments are congruent.

So the green segments are both 10.

The brown segments

start from the same point *C*

and are tangent to the same circle.

So the brown segments are congruent.

So the brown segments are both 5.

So the perimeter of △*ABC* is*P* = 2(7 + 10 + 5).

7 + 10 = 17

17 + 5 = 22

So *P* = 2⋅22.

2⋅22 = 44

So *P* = 44.