In the circle below, , , and . Find .
Secant Lines to Circles
Recall that a line that intersects a circle in exactly one point is called a tangent line. A line that intersects a circle in two points is called a secant line. Below, is a secant.
When two secants or a tangent and a secant are drawn, they can interact in four ways. In each case, arcs, angles and line segments have special relationships. These ideas are summarized below, and will be explored further and proved in the examples and practice.
Case #1: Two secants intersect outside the circle.
Relevant Theorems:
- (This will be explored in #1 below)
- (This will be explored in #2 below)
Case #2: Two secants intersect inside the circle.
Relevant Theorems:
- (This was previously proved as a property of intersecting chords)
- (This will be explored in #3 below)
Case #3: A secant and a tangent intersect on the circle.
Relevant Theorem:
- (This will be explored in Example 1)
Case #4: A secant and a tangent intersect outside the circle.
Relevant Theorems:
- (This will be explored in the Review problems)
- (This will be explored in the Review problems)
Let's take a look at some problems involving secant lines.
1. Prove that .
Draw chords and .
Two triangles are created, and . Note that both triangles share . Also note that both and are inscribed angles of . Therefore, . Because and have two pairs of congruent angles, they are similar by . This means that corresponding sides of the triangles are proportional. In particular, . This means that .
2. Prove that .
You are trying to prove that the measure of the angle is equal to half the difference between the measures of the red arc and the blue arc. As in #1, draw chords and .
Consider how the angles are arcs are related.
- (inscribed angle)
- (inscribed angle)
- (exterior angle equals the sum of the remote interior angles)
Make two substitutions and you have:
Therefore,
3. Prove that .
This logic of this proof is similar to the logic used in #2. Start by drawing chord .
Consider how the angles and arcs are related.
- (inscribed angle)
- (inscribed angle)
- (exterior angle equals the sum of the remote interior angles)
Make two substitutions and you have:
Therefore, . Because and are vertical angles, they are congruent and have equal measures. This means .
Examples
Example 1
Earlier, you were given a problem about a secant line to a circle.
In the circle below, , , and . Find .
This is an example of two secants intersecting outside the circle. The intersection angle of the two secants is equal to half the difference between their intercepted arcs. In other words, . You are given , but you don't know . Use the fact that a full circle is to find .
Now, solve for the measure of .
Example 2
is tangent to circle at point . Prove that .
Draw a diameter through points and . This segment will be perpendicular to .
First note that because the two arcs make a semicircle. This means that and thus .
Now consider other angle and arc relationships:
- (inscribed angle)
- (two angles make a right angle)
By substitution, . Therefore, .
Consider the two highlighted statements. Both and are equal to . Therefore, .
Example 3
. Find .
If , then . Therefore, .
Example 4
and . Find .
and . is the average of the measure of the intercepted arcs.
Therefore, .
Review
1. What's the difference between a secant and a tangent?
Use the relationships explored in this concept to solve for or in each circle.
2.
3.
4.
5.
6.
7.
In #8-#12 you will explore Case #4: A secant and a tangent intersect outside the circle.
8. Draw chord . Explain why .
9. Prove that .
10. Prove that .
11. Prove that (Use Example B to help).
12. Prove that .
13. How is the theorem proved in #11-#12 related to the theorem proved in Examples B?
Solve for or in each circle.
14.
15.
Review (Answers)
To see the Review answers, open this PDF file and look for section 8.8.