Tangent to a Circle

How to use the properties of the tangent to the circle: definition, 2 properties, 2 examples, and their solutions.

Definition

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

Property

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

Example 1

Example

Solution

Ray PB is tangent to the given circle
at point B.
And OB is the radius.

So ray PB and OB are perpendicular.

OB is the radius.
OB = 5.

OA is also the radius.

So OA = 5.

See △OPB.
It's a right triangle.

The sides are (5, PB, 8 + 5) = (5, PB, 13).

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

Pythagorean Triple

So PB = 12.

So write 12.

So 12 is the answer.

Property 2

Property

If two segments from an exterior point
are tangent to a circle,

then those two segments are congruent.

Example 2

Example

Solution

The blue segments
start from the same point A.

They are tangent to the same circle.

So the blue segments are congruent:
7.

The green segments
start from the same point B.

They are tangent to the same circle.

So the green segments are congruent:
10.

The brown segments
start from the same point C.

They are tangent to the same circle.

So the brown segments are congruent:
5.

See △ABC.

There are 2 of 7, 10, and 5.

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

7 + 10 + 5 = 22

2⋅22 = 44

So 44 is the answer.