Tangent to a Circle

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. A tangent and a circle have one intersecting point. From a point that is exterior of the circle, you can draw two tangents.

A tangent to a circle is a line
that touches the circle.

A tangent and a circle have one intersecting point.

From a point that is 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.

The tangent and the radius are perpendicular
at the intersecting point of the circle.

Example 1

Ray AB is tangnet to the given circle at point B. Find the value of x. Radius: 5, AB: x, The distance between point A and the other point on the circle: 8.

The radius is 5.

So the distance between point A
and the center of the circle is 8 + 5, 13.

The blue tangent and the radius
are perpendicular at point B.

So the given triangle is a right triangle.

The hypotenuse is 13.
And the leg is 5.

So the triangle is a (5, 12, 13) right triangle.

So x = 12.

Property 2

If two segments from an exterior point are tangent to a circle, then those two segments are congruent.

If two segments from an exterior point
are tangent to a circle,
then those two segments are congruent.

Example 2

Find the perimeter of triangle ABC. The lengths of the tangent segment: 7, 10, 5.

The same colored segments
start from the same point,
and are tangent to the circle.

So the same colored segments are congruent.

So the perimeter P is:
P = 2(7 + 10 + 5).

This is the way to find the perimeter
of the circumscribed polygon.