Unit 11
Segments and Equations of Circles
Lesson 1: Properties of Tangents
Opening Exercise
Draw 3 different diagrams of a circle and a line given the following:
They do NOT intersect. / They intersect once. / They intersect twice.A line that intersects a circle at exactly two points is called a ______line.
A line that intersects a circle at exactly one point is called a ______line.
Example 1
You will need a protractor.
In the accompanying diagram, P is called:
Using a protractor, measure the angle formed by the radius and the tangent line. Write the angle measure on the diagram.
Will this work for all angles formed by a radius and a tangent line?
Important Discovery!
A tangent line to a circle is ______to the radius of the circle drawn to the point of tangency.
The converse is also true. So, a line through a point on a circle is tangent at the point if, and only if, it is perpendicular to the radius drawn to the point of tangency.
4 Common Tangents(2 completely separate circles)
/ 3 Common Tangents
(2 externally tangent circles)
/ 2 Common Tangents
(2 overlapping circles)
1 Common Tangent
(2 internally tangent circles)
/ 0 Common Tangents
(2 concentric circles)
Concentric circles are circles
with the same center.
/ (one circle floating inside the other, without touching)
Tangent lines that meet two circles are called ______tangents. Listed below are the 5 different ways we can discuss common tangents.
Example 2
In the diagram, and are tangent to circle A at points D and E respectively.
Write a two-column proof to prove .
Statements Reasons
1. and are tangent to circle A at 1. Given
points D and E respectively
2. and are right angles 2.
3. and are right triangles. 3.
4. 4.
5. 5.
6. 6.
7. 7.
Example 3
In circle A, the radius is and .
a. Find .
b. Find . Explain how you know.
c. Find . Explain how you know.
d. Find the area of .
e. Find the perimeter of quadrilateral .
Example 4
If , , and , is tangent to circle A at point B? Explain.
Example 5
You will need a compass and a ruler.
Construct a line tangent to circle A through point B.
Exercises
1. In the diagram, circle O is inscribed in so that the circle is tangent to at F, to at E, and to at D.
If and , find the perimeter of .
2. In circle A, , , and .
a. Find the length of the radius of the circle.
b. Find (to the nearest tenth).
c. Find
Homework
1. If , , and , is tangent to circle A? Explain.
2. In the given figure, the three segments are tangent to the circle at point F, B and G. Find DE.
3. In the given figure, circles X and Y have two tangents drawn to them from external point T. The points of tangency are C, A, S, and E. The ratio of TA to AC is 1:3.
If TS =24, find the length of SE.
Lesson 2: Tangent Segments and Angles
Opening Exercise
Find x if the line shown is tangent to the circle at point B.
Example 1
Given circle A with tangent .
a. Draw . What is the measure of ? Explain.
b. What is the measure of ? Explain.
c. Express the measure of the remaining two angles of in terms of a. Explain.
d. What is the measure of in terms of a? Show how you calculated your answer.
e. Summarize what we have just proven.
Exercises
1. Solve for a. 2. Solve for a.
3. Solve for a.
We have learned a lot about tangents! Here is a summary:
· A tangent line intersects a circle at exactly one point (and is in the same plane).
· The point where the tangent line intersects a circle is called the point of tangency.
· The tangent line is perpendicular to a radius whose endpoint is the point of tangency.
· The two tangent segments to a circle from an exterior point are congruent.
· The measure of an angle formed by a tangent segment and a chord is one-half the angle measure of its intercepted arc.
· If an inscribed angle intercepts the same arc as an angle formed by a tangent segment and a chord, then the two angles are congruent.
Example 2
Find the values of a, b, and c.
Example 3
Find the values of a, b, and c.
Homework
1. Calculate the value of z. 2. Find the values of a and b.
3. Complete the following two-column proof.
Given: Circle P with tangents AC and AB
Ray AP is drawn
Prove: bisects
Statements Reasons
1. Circle P with tangents AC and AB 1. Given
Ray AP is drawn
2. Draw radii BP and CP 2. Auxiliary Lines
3.
Lesson 3: Interior and Exterior Angles
Opening Exercise
Vocabulary
Definition / DiagramSecant Line
· a line that intersects a circle in exactly two points /
What is the difference between a tangent and a secant?
On the given circle, draw two secants that:
a. intersect inside the circle. b. intersect outside the circle.
c. intersect on the circle. d. do not intersect.
Example 1
Using our knowledge of inscribed angles, we are going to find the measure of an interior angle that is not a central angle.
To find x, draw chord BD.
Can you determine any of the angle measures in ? Explain.
Find x. Justify your answer.
Interior Angle (vertex inside the circle)
Formula
Example 2
a. Find the value of x: b. Find the value of x:
Exercises
1. Find the values of angles x and y. 2. Find the value of x.
Exterior Angle (vertex outside the circle)
Formula
Example 3
Write the equation used to find in the following diagrams:
Exercises
1. Find the measure of .
2. Find the measure of .
Homework
1. Find the value of x. 2. Find the measure of .
3. Find the measure of .
Lesson 4: Interior and Exterior Angles II
Opening Exercise
Find the value of x in the diagrams pictured below:
Example 1
Find the value of x.
Example 2
If , solve for x.
Example 3
Find the values of x and y.
Exercises
In the following questions, find the value of x:
1.
2.
Homework
1. Find the values of x and y.
2. Find the value of x.
3. Find .
Lesson 5: Similar Triangles in Circle–Secant Diagrams
Opening Exercise
Given: Circle with chords and
intersecting at point F
Prove:
Exercises
1. Find the value of x.
2. In the circle shown, , , and . Find the shorter part of .
Segment Lengths
It is also true that when secant lines, tangent lines, or secant and tangent lines intersect outside of a circle, their segment lengths can be found using:.
In words, .
Sometimes the whole and the outside piece are one in the same. In this case, the formula is .
In words, .
Exercises
1. Find the value of x in simplest radical form.
2. If , , and , find .
3. Find the value of x.
Homework
1. Find the value of x.
2. Find the value of x.
3. Find the value of x.
Lesson 6: Writing the Equation of a Circle
Opening Exercise
a. Find the length of the line segment shown on the coordinate plane below.
b. Using the distance formula, find the distance between the points and .
Example 1
If we graph all of the points whose distance from the origin is equal to 5, what shape will be formed?
Using the given coordinate plane, plot 4 points that are 5 units away from the origin.
Now, we need to find 4 more.
Write down any ideas that you might have to find the location of the next point that is also 5 units from the origin.
Compare your plan with a partner. Once you agree on a plan, plot three more points using this method. Using your compass, connect these points to form a circle.
In the above circle, the center is located at ______and the radius length is ______.
We found the location of a point on the circle by using
______, which states ______.
If we generalize this formula by using a point named , the point will satisfy the equation when the circle has a center at the origin.
Example 2
Now, let’s look at a circle that is not centered at the origin.
This circle is centered at and has a radius length of 5 units.
Is this circle congruent to the circle we constructed?
Is there a sequence of basic rigid motions that would take this circle center to the origin? Explain.
The equation for this circle can be found using this same pattern to move the center of the circle back to the origin.
The equation of this circle is:
Example 3
Write the equation of the circle that is graphed below.
Example 4
Find the radius and center of the circle given by the equation:
Example 5
Write an equation for the circle whose center is at and has radius 7.
Homework
1. Describe the circle given by the equation: .
2. Write the equation for a circle with center and radius 8.
3. Write the equation for the circle shown.
Lesson 7: Writing the Equation of a Circle II
Opening Exercise
Two points in the plane, and , represent the endpoints of the diameter of a circle.
a. What is the center of the circle? Explain.
b. What is the radius of the circle? Explain.
c. Write the equation of the circle.
Example 1
Write the equation of a circle with center that passes through ?
Example 2
A circle with center is tangent to the x-axis.
a. What is the radius of the circle?
b. What is the equation of the circle?
Example 3
Given a circle centered at the origin that goes through point (0, 2), determine whether or not this circle would go through the point .
Example 4
Determine the center and radius of each circle:
a.
b.
c.
Homework
1. Determine the center and radius of the circle .
2. Write the equation of a circle that has a center of (-4, -3) and is tangent to the y-axis.
3. A circle has a diameter with endpoints at and . Write the equation for this circle.
Lesson 8: Recognizing Equations of Circles
Opening Exercise
Complete the following table:
Polynomial / Factored FormExample 1
Find the center and the radius of the following:
a.
b.
Example 2
Find the center and the radius of the following:
Example 3
Could the circle with the equation have a radius of 4? Why or why not?
Example 4
Identify the graphs of the following equations as a circle, point, or an empty set.
a. b.
Summary
When is … / The figure is …Positive
Negative
Zero
Exercises
1. The graph of the equation below is a circle. Identify the center and radius of the circle.
2. Identify the graphs of the following equations as a circle, point, or an empty set.
a.
b.
c.
Example 5
Chante claims that two circles given by and are externally tangent. She is right. Show that she is.
Homework
1. Identify the center and radius of the following circles.
a.
b.
c.
d.
2. Sketch a graph of the equation .
Lesson 9: Inscribed and Circumscribed Circles
Opening Exercise
In each diagram, try to draw a circle with center D that is tangent to both rays of .
Which diagrams did it seem impossible to draw such a circle? Why did it seem impossible?
What do you conjecture about circles tangent to both rays of an angle? Why do you think that?
Important Discovery!
If a circle is tangent to both rays of an angle, then the center of the circle lies on the:
Example 1
You will need a compass and a straightedge.
Construct a circle that is tangent to both rays of the given angle.
1. How do you find the center?
2. How do you find the radius?
Now let’s make the construction!
Example 2
You will need a compass and a straightedge.
Let’s construct a circle inscribed in a triangle!
In the space below, using a straightedge, draw a large triangle.
a. Pick any two angles and construct their angle bisectors.
b. What is special about the intersection point of these angle bisectors?
c. Construct a perpendicular segment from this intersection point to any side of your triangle. What is this segment called?
d. Using your compass, the intersection point of your angle bisectors, and this segment length, construct a circle. This is called the incircle.
We have now discussed points of concurrency in triangles over the course of the year. Let’s take a look at them one more time to see how this relates to inscribed and circumscribed circles.
Draw in the points of concurrency in the diagrams below:
Centroid / Incenter / Circumcenter / Orthocentermedians / angle bisectors / perpendicular
bisectors / altitudes
Example 3
You will need a compass and a straightedge.
Construct a circle so that it is circumscribed around the triangle pictured. This is called the circumcircle.
Recall: To find the inscribed circle, we used incenter.
To find the circumscribed circle, we will use ______.
Exercises