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UNIT 7: CIRCLES
Circle Basics
Definition
A circle is
The radius is_________________________________________________
Other terms
Name:
Congruent Circles have the same
Concentric circles
.
Internally Tangent Circles
Area and Circumference of a Circle
Externally Tangent Circles
UNIT 7: CIRCLES
Tangent to a Circle Theorem:
Example 1: In A, BC is tangent at point B.
a. Find AC. Round to the hundredth.
b. Find DC. Round to the hundredth.
Example 2: Using the triangle, determine if BC is
tangent to the circle at point B.
Example 3: CD is tangent to A and B at
points D and C, respectively. Find the distance
between the centers of the two circles. Round to
the tenth.
Ice Cream Cone Theorem
Example 4: AB , BC , and AC are tangent to
at points E, F, and D, respectively. Find the
perimeter of ABC.
G
Circumscribed Angle Theorem
Example 6: In Example 5, if mB  42 , find mADC .
Example 5: Find the value of x.
UNIT 7: CIRCLES
Circles on the Coordinate Plane
The equation of a circle
Recall that a circle is the set of all points that are a fixed distance (radius) from fixed point
(center). Using the distance formula,
r
 x  h   y  k 
2
2
By squaring both sides, this gives us the general equation of a circle on the coordinate plane.
r 2   x  h   y  k 
2
2
where r is the radius of the circle, and (h, k) is the center.
Example 7: Write the equation of the circle with a center (-2,5) and radius of 4 units.
 x  h  y  k   r 2
2
2
 x  -2    y  5   42
2
2
 x  2    y  5   16
2
2
Example 8: If the equation of a circle is  x  1   y  6   10 , what is the center and radius?
2
2
 x  h  y  k   r 2
2
2
 x  -1   y  6   10
2
2
Center (-1,6) and Radius =
10 units
Example 9: If the center of a circle is (2,3) and the point (-4,-1) is on the circle, write the equation of the circle.
62  42 =
Distance between center and point on circle is the radius:
52
 x  h  y  k   r 2
2
2
 x  2    y  3   52
2
2
Example 10: If the equation of a circle is  x  1   y  6   45 , is the point  7,3  on the circle? Support
your answer.
2
2
Plug in point (-7,3) for (x, y) in the equation:
 -7  1   3  6   45
2
2
 6    3   45
2
2
36  9  45
45  45
Because it makes equation true, the point  7,3  is on the circle.
UNIT 7: CIRCLES
Proving that Two Circles are Similar
Two figures are similar if and only if there is a sequence of similarity transformations that maps one figure
onto the other.
Example 11: Prove that the circles are similar by identifying a sequence of
similarity transformations that will map one circle onto the other.
Angles and Arc Measure
Example 12: Find the following measures in
A . BE is a diameter.
1. mCD =
2. mDEF =
3. mDFC =
4. mEAD =
5. Are there semicircles in the figure? If so, name them.
Congruent Circle Parts Theorem
Example 13: In
A , mBD  125 , what is mCD ?
UNIT 7: CIRCLES
Example 14:
C  J and mDCG  mNJM .
Find DG.
Radius to a Chord Theorem
If a radius (or diameter) is perpendicular to a chord, it bisects the chord and its intercepted arc.
If a line is perpendicular to and bisects a chord, then it contains the radius (or diameter) of the
circle.
Example 15: In
R , RS  8 , and SM  9 .
Find the length of the radius of the circle.
Find SN and NP.
Find mNP rounded to the nearest tenth.
Inscribed Angles
An inscribed angle is an angle with
.
Inscribed Angle Theorem
An inscribed angle is
the measure of its intercepted arc.
UNIT 7: CIRCLES
Semicircle Corollary
An inscribed angle intercepting a semi-circle is
Example 16: In
.
O , PM is a diameter. Find the following.
mPN =
mMN =
mLNP =
mMPN =
mLPN =
mMNP =
mLM =
mPMN =
Inscribed Quadrilateral Theorem
An inscribed polygon is a polygon placed inside a circle, so that
A quadrilateral inscribed in a circle has opposite angles that are
Example 17: Each quadrilateral is inscribed in the circle. Find the value of the variables.
Example 18: Quadrilateral ABCD is inscribed in the circle. Find the all of the angle measures of
quadrilateral ABCD.
.