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. 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.

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.

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.

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.

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 segments: 7, 10, 5.

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.