Name: _________________________________________________
Mrs. White
Date
Lesson
Topic
5/4
1
Graphs and Equations of Circles
5/5
2
Completing the Square with Circle Equations
5/6
3
More Equations of Circles
5/7
4
Area of a Sector and Arc Length
5/8
5
Similar Circles and Radian Measure
5/11
6
Mixed Practice
QUIZ
5/12
7
Circle Proofs Revisited
5/13
8
More Practice with Circle Proofs
5/14
9
Unit 13 Review
5/15
10
Unit 13 Test
Page 1 of 22
Lesson 1 - Graphs and Equations of Circles
Remember the distance formula? _______________________________
All of the points on a circle are __________________________________________________________.
As a result, The equation of a circle is based on the ____________________ _________________

Let the center of a circle be the point (h, k) and
let the point (x, y) represent any point that is
located on the circle. (See diagram)

Use the distance formula to write an equation
to represent the value of r.
This is where the equation of a circle comes from:
The equation of a circle in Standard Form with center (___, ___) and radius ___ is:
Ex 1: Find the radius and the center of each circle:
Radius
Center
a. (x – 6)2 + y2 = 25
_________
_____________
b. x2 + (y + 2)2 = 49
_________
_____________
c. (x + 13)2 + (y + 5)2 = 81 _________
_____________
e. x2 + y2 = 1
_________
_____________
f. x2 + y2 = 18
_________
_____________
Ex 2: Write an equation of the circle with the following centers/radii:
a. C(0,5), r = 6
b. C(-3,4), r =
5
Page 2 of 22
Ex 3: Write the equation of the circle with
center at the origin and radius of 4.5.
Ex 4: Write the equation of the circle with
center (4,-1) that passes through (5, 2)
Review: Midpoint Formula: _________________
Ex 6:
Write the equation of the circle whose
diameter has endpoints (4, -3) and (-2, 5).
Ex 7:
Write the equation of the circle whose
diameter has endpoints (4,-1) and (-6, 7).
Graphing a Circle
To graph a circle:
Step 1: Find the center and radius AND LIST THEM
Step 2: Graph the circle
Ex. 1: x2 + y2 = 5
Page 3 of 22
Ex. 2: Graph the following equations:
C. (x - 3)2 + (y + 4)2 = 4
Ex. 3: Write the equations of the following circles
Page 4 of 22
Lesson 2: Completing the square and Equations of Circles
Warm-Up:
Find the radius and the center of each circle:
a) x2 + (y+3)2 = 36
b) (x-1)2 + (y-4)2 = 32
Write the equation of a circle with center (6, -8) and radius 7 .
Writing the Equation of a Circle in General Form
Standard form of a circle is ____________________________________
General Form of a Circle is 𝐴𝑥 2 + 𝐵𝑦 2 + 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0
Examples: 𝑥 2 + 𝑦 2 + 2𝑥 + 3𝑦 + 5 = 0
2𝑥 2 + 2𝑦 2 + 𝑥 − 𝑦 − 7 = 0
Converting from Standard Form to General Form of a circle:
Example: Convert each question to General form.
a) 𝑥 2 + 𝑦 2 = 25
b) 𝑥 2 + (𝑦 + 2)2 = 9
c) (𝑥 − 3)2 + (𝑦 − 4)2 = 16
TRY IT!! Given the center (-3, 0) and radius r = 2, write the equation of the circle in general
form.
Page 5 of 22
Converting from General Form of a circle to Standard Form
This process is more complicated, but it is also more useful because in standard form, you
can easily see the ____________________________________________________ !
Review of completing the square:
 Get constant term isolated to one side and leading coefficient must = 1.
 Find a number to add to the quadratic and linear term in order to make a perfect
square trinomial (i.e. cut linear coefficient in half and square it)
Example: Solve for x by completing the square.
a) 𝑥 2 + 2𝑥 − 8 = 0
b) 2x2 + 28x - 30 = 0
You try one:
c) 𝑥 2 − 10𝑥 − 11 = 0
Converting from General Form of a circle to Standard Form:
 Complete the square for BOTH x AND y
 This will be shorter than fully completing the square since you are not solving for a
variable
Example:
a) 𝑥 2 + 𝑦 2 + 2𝑥 + 4𝑦 − 11 = 0
Page 6 of 22
b) 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 4 = 0
What is the center and radius of this circle?
Find the center and radius of hte following circles:
c) 𝑥 2 + 𝑦 2 + 6𝑥 + 5 = 0
d) 𝑥 2 + 𝑦 2 + 8𝑥 − 2𝑦 − 8 = 0
e) 𝑥 2 + 𝑦 2 = 10𝑦 + 24
Page 7 of 22
Lesson 3: More Equations of Circles
Warm-up:
1) Write the equation of a circle with radius r whose center is (h, k).
2) Write the equation of a circle with radius r whose center is at the origin?
Write the equation of a circle whose diameter has the given endpoints.
1) (1, 4) and (-3, 4)
2) (5, 4) and (0, -8)
3.
Page 8 of 22
Sketch the circle.
Convert to General form.
a) (𝑥 + 2)2 + (𝑦 − 6)2 = 16
b) (𝑥 − 1)2 + (𝑦 − 1)2 = 4
Page 9 of 22
Graph the following equations by first writing them in standard form. Be sure to list the
center and radius of each first.
a) 𝑥 2 + 𝑦 2 + 8𝑥 + 4𝑦 + 19 = 0
b) 𝑥 2 + 𝑦 2 − 10𝑦 + 9 = 0
c)
𝑥 2 + 𝑦 2 − 10𝑥 = 24
Be careful of scale!
Make sure your graph
fits on your grid!
Page 10 of 22
Lesson 4: Sector Area and Arc Length
Warm Up: 1. Name or find the following:

̂
2. If mAOB  56 , find 𝐴𝐵
Center: __________
A
Radius: __________
Diameter: __________
D
Arc: __________
O
B
Central Angle: __________
Area (formula): __________
C
Circumference (formula): __________
Area of a Sector
A _____________ of a circle is a region bounded by two _________ of the circle and their
intercepted arc.

Sector _____ is illustrated to the right.

The ______ of a sector is a fraction of the circle’s area.
The area of the sector depends upon how much of the circle is taken up by the sector. As a
result, we can it to the measure of the arc of the sector.
We can write and solve the following proportion to find the area of a sector:
HELPFUL HINT: Write the degree symbol after m in the formula to help you remember
to use degree measure NOT arc length (more on this in a little bit).
Page 11 of 22
TRY IT!
1)
2)
Arc Length
An ______________ of a circle is the distance along an ________ measured in _____________
(such as __________________________________________), NOT degrees.

The arc length is a fraction of the circle’s ________________________.
We can set up a similar proportion to find arc length, except we will compare the arac length
to the circumference instead of the area.
Finding Arc Length
Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth.
Page 12 of 22
TRY IT!
Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth.
1)
2) An arc length with measure 135 degrees in a circle with radius 4 cm.
Find the area of the dark blue sectorshown at the left.
The radius of the circle is 4 units and the length of the
arc (the curved edge of the sector) measures 7.85 units.
Express answer to thenearest tenth of a square unit.
Summary:
What is the difference between arc measure and arc length?
Write the formula for the Area of a Sector
Write the formula for the Length of an Arc
Page 13 of 22
Lesson 5: Similar Circles and Radian Measure
Warm Up:
1. In the accompanying diagram, the center of circle O is
coordinates of point P are
circle?
. If
, and the
is a radius, what is the equation of the
3. The center of a circular sunflower with a diameter of 4 centimeters is
What equation represents the sunflower?
.
Similar Circles
In previous units we defined two figures to be
similar if there was a single or sequence of similar
transformations that could map the one figure
onto the other.
The similarity transformations consist of
translations, rotations, reflections and dilations.
These all preserve the shape of the figure.
Due to unique nature of a circle – that it is
defined by a single length, the radius, it seems
obvious that we could use a scale factor to alter the radius in such a way that it could be the
same size as another circle.
Using a composition of transformations involving a translation followed by a dilation would
always work to establish two circles to be similar. A translation would form two concentric
circles (circles with the same center). Once the circles share a common center then from that
center point we could perform a dilation of R:r or r:R… both of
these scale factors would map one circle onto the other.
Using the scale factor R:r would be a reduction and would map
the large circle onto the smaller circle. Using the scale factor r:R
would be an enlargement and would map the small circle onto the
large circle.
Either of these establish that using only similarity
transformations, a translation and a dilation, we could map any
circle onto another….
THUS ALL CIRCLES ARE SIMILAR.
Page 14 of 22
Radian Measure
A ______________ is a unit of measure for ________________.
But what is a radian??
One radian is the angle made at the center of a circle by an
__________ whose length is equal to the __________ of the circle.


As seen in the figure above, a radian is defined by an arc of a
circle. The length of the arc is equal to the radius of the circle.
Even though circles vary in size, they are ____________________
and as a result, a radian will ALWAYS be the same size. (Just
like one inch is always the same).
How many radians (radii) are in a full circle?


Since a radian is based on the radius placed along the circle, we should look at
___________________________
A full circle has circumference of C = 2𝜋r so number of radians (radii) = _______=
Example:
If a circle has a radius of 5, then number of radians is ______ = _____.
We are used to degrees, so it would be helpful to convert from one to the other.
So we know that a full circle has ___________ or ______ radians.
We can use this information to convert back and forth from radians to degrees.
Solve for 1o:
360o = 2𝜋 radians
Solve for one radian:
360o = 2𝜋 radians
To convert from radians to degrees, multiply by ___________________________
To convert from degrees to radians, mutliply by _____________________________
Page 15 of 22
Example: Let’s convert 30 to radians.
Radians =
𝜋
(180) (30°) = ____________ radians
Now you try. Convert the following to radians.
1. 45
2. -60
Example: Let’s convert
degrees =
3. 90
7
to degrees.
6
180 7 

= _____

6
Now you try. Convert the following radians to degrees.
1.
11
6
2.
5
4
3. 2.7
*Hint: to convert to radians, you usually want π in your answer so multiply by 𝝅⁄𝟏𝟖𝟎 *
Mixed Practice:
1. Determine the translation vector that would map the center of circle A onto the center of
circle B.
a)
b)
Circle A
Circle B
Circle A
Circle B
A (2, 2)
B (8, 2)
A (0, -9)
B (-4, -6)
Page 16 of 22
2. What scale factor would make circle A the same size as circle B?
a)
b)
Circle A
Circle B
Circle A
Circle B
RadiusA =
8cm
RadiusB =
1cm
RadiusA = 8cm
RadiusB =
12cm
Scale Factor: _____
Scale Factor: ______
Scale Factor: _____
3. Name the series of transformations that would map circle A onto circle B.
a)
b)
B
c)
A
B
A
A
B
3. Convert the following radian measures to degrees.
a.
3
4
b.
5
6
4. Convert the following degree measures into radians.
a. - 120
b. 75
Page 17 of 22
Lesson 6: Mixed Practice
Warm Up:
1. An arc is cut from a circle. The circle has a radius of 2.5 cm and the arc has a measure of
3
𝜋 radians. Find, to the nearest hundredth of a cm, the length of the arc.
4
2. Which equation of a circle will have a graph that lies entirely in the first quadrant?
1)
2)
3)
4)
3. List center and radius of the circle with the equation x2 + y2 + 6y - 4 = 0.
4. Convert the following degree measures
to radians. Leave in terms of pi.
5. Convert the following radian measures
to degrees.
a. 50
a.
b. 120
𝜋
2
b.
5𝜋
4
6. A circle has a radius of 5. A sector is cut from the circle that has an arc measure of 40°.
Find to the nearest tenth the area that is remaining in the circle.
Page 18 of 22
Lesson 7: Circle Proofs Revisited
Warm Up:
 If two inscribed angles intercept the same or congruent arcs, then ___________________
_____________________________________________________________________________________
 Two chords are congruent if and only if the arcs they intercept are ___________________
 In a circle cut by parallel lines, ______________________________________________________
 If an angle is inscribed in a semicircle, then _________________________________________
 If a diameter/radius is perpendicular to a chord then, _______________________________
_____________________________________________________________________________________
 A line is __________________________________________________ if and only if it is
perpendicular to the radius of the circle at the point of tangency.
 An inscribed angle is _________ the measure of the ____________________________
 A central angle is equal to the measure of ________________________________________
 Name the most common way to prove triangles similar: ________________
 If two triangles are similar, then ____________________________________________________
 In a proportion, the product _______________________________________________________
1.
Page 19 of 22
Proving a Product:
Method:
1. Determine the triangles to prove similar by marking up your diagram
 Highlight the segments in one product in one color and the segments in the
other product in another color.
 The triangles should include one side of each color.
2. Begin youir proof by finding the two pairs of angles that show the triangles are similar.
Page 20 of 22
Lesson 8: More Circle Proofs
Warm Up:
For example 1 below, determine which triangles we need to prove similar.
Example 1:
Page 21 of 22
Example 2:
Example 3:
Chords
and
of circle O intersect at E,
an interior point of circle O;
Chords
and
are drawn.
Prove: (AE)(EB) = (CE)(ED)
C
A
O
E
D
B
Page 22 of 22