Title: READ: Tangent Lines Reading
Course: Geometry (B)

Tangent to a Circle

Tangent to a Circle Theorem:

A tangent line is always at right angles to the radius of the circle at the point of tangency.

Proof. We will prove this theorem by contradiction.

We start by making a drawing. is a radius of the circle. is the center of the circle and is the point of intersection between the radius and the tangent line.

Assume that the tangent line is not perpendicular to the radius.

There must be another point on the tangent line such that is perpendicular to the tangent line. Therefore, in the right triangle is the hypotenuse and is a leg of the triangle. However, this is not possible because . (Note that ).

Since our assumption led us to a contradiction, this means that our assumption was incorrect. Therefore, the tangent line must be perpendicular to the radius of the circle.

Since the tangent to a circle and the radius of the circle make a right angle with each other, we can often use the Pythagorean Theorem in order to find the length of missing line segments.

Example 1

In the figure, is tangent to the circle. Find .

Since is tangent to the circle, then .

This means that is a right triangle and we can apply the Pythagorean Theorem to find the length of .

Example 2

Mark is standing at the top of Mt. Whitney, which is tall. The radius of the Earth is approximately (There are in one mile.) How far can Mark see to the horizon?

We start by drawing the figure above.

The distance to the horizon is given by the line segment .

Let us convert the height of the mountain from feet into miles.

Since is tangent to the Earth, is a right triangle and we can use the Pythagorean Theorem.

Converse of a Tangent to a Circle

Converse of a Tangent to a Circle Theorem

If a line is perpendicular to the radius of a circle at its outer endpoint, then the line is tangent to the circle.

Proof.

We will prove this theorem by contradiction. Since the line is perpendicular to the radius at its outer endpoint it must touch the circle at point . For this line to be tangent to the circle, it must only touch the circle at this point and no other.

Assume that the line also intersects the circle at point .

Since both and are radii of the circle, , and is isosceles.

This means that .

It is impossible to have two right angles in the same triangle.

We arrived at a contradiction so our assumption must be incorrect. We conclude that line is tangent to the circle at point .

Example 3

Determine whether is tangent to the circle.

is tangent to the circle if .

To show that is a right triangle we use the Converse of the Pythagorean Theorem:

The lengths of the sides of the triangle satisfy the Pythagorean Theorem, so is perpendicular to and is therefore tangent to the circle.

Tangent Segments from a Common External Point

Tangent Segments from a Common External Point Theorem

If two segments from the same exterior point are tangent to the circle, then they are congruent.

Proof.

The figure above shows a diagram of the situation.

- Given: is a tangent to the circle and is a tangent to the circle
- Prove:

Statement | Reason
1. is tangent to the circle | 1. Given
2. | 2. Tangent to a Circle Theorem
3. is a tangent to the circle | 3. Given
4. | 4. Tangent to a Circle Theorem
5. | 5. Radii of same the circle
6. | 6. Same line
7. | 7. Hypotenuse-Leg congruence
8. | 8. Congruent Parts of Congruent Triangles are Congruent

Example 4

Find the perimeter of the triangle.

All sides of the triangle are tangent to the circle.

The Tangent Segments from a Common External Point Theorem tells us that:

Example 5

An isosceles right triangle is circumscribed about a circle with diameter of . Find the hypotenuse of the triangle.

Let’s start by making a sketch.

Since and are radii of the circle and and are tangents to the circle,

and .

Therefore, quadrilateral is a square.

Therefore,

We can find the length of side by using the Pythagorean Theorem.

and are both isosceles right triangles, therefore and all the corresponding sides are proportional:

We can find the length of by using one of the ratios above:

Cross-multiply to obtain:

Rationalize the denominators:

The length of the hypotenuse is

Corollary to Tangent Segments Theorem

A line segment from an external point to the center of a circle bisects the angle formed by the tangent segments starting at that same external point.

Proof.

- Given:
  - is a tangent to the circle
  - is a tangent to the circle
  - is the center of the circle

- Prove

Proof.

We will use a similar figure to the one we used we used to prove the tangent segments theorem (pictured above).

by congruence.

Therefore, .

Example 6

Show that the line is tangent to the circle . Find an equation for the line perpendicular to the tangent line at the point of tangency. Show that this line goes through the center of the circle.

To check that the line is tangent to the circle, substitute the equation of the line into the equation for the circle.

This has a double root at . This means that the line intersects the circle at only one point .

A perpendicular line to the tangent line would have a slope that is the negative reciprocal of the tangent line or .

The equation of the line can be written: .

We find the value of by plugging in the tangency point:

The equation is and we know that it passes through the origin since the intercept is zero.

This means that the radius of the circle is perpendicular to the tangent to the circle.

Common External Tangents

Here is an example in which you might encounter the use of common external tangents.

Example 1

Find the distance between the centers of the circles in the figure.

Let’s label the diagram and draw a line segment that joins the centers of the two circles. Also draw the segment perpendicular the radius

Since is tangent to both circles, is perpendicular to both radii: and .

We can see that is a rectangle, therefore .

This means that .

is a right triangle with and . We can apply the Pythagorean Theorem to find the missing side, .

The distance between the centers is approximately

Lesson Summary

In this section we learned about tangents and their relationship to the circle. We found that a tangent line touches the circle at one point, which is the endpoint of a radius of the circle. The radius and tangent line are perpendicular to each other. We found out that if two segments are tangent to a circle, and share a common endpoint outside the circle; the segments are congruent.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

Review Questions

1. Determine whether each segment is tangent to the given circle:
2. Find the measure of angle .
3. Find the missing length:
4. Find the values of the missing variables
5. Find the perimeter of the pentagon:
6. Find the perimeter of the parallelogram:
7. Find the perimeter of the right triangle:
8. Find the perimeter of the polygon:
9. Draw the line and the circle . Show that these graphs touch at only one point.

Find the slope of the segment that joins this point to the center of the circle, and compare your answer with the slope of the line .

Review Answers

  1. Yes
  2. Yes
  3. No
9. solve for to obtain double root .

The slope of the line from to , which is the negative reciprocal of the slope of the line.

This means that the tangent line and radius are perpendicular.