MM2G3b,d
Central and Inscribed Angles of Circles
Question: What are some of the properties of chords and angles in a circle?
Launch: Can you think of any reason we may need to know the circumference, diameter, and/or
the radius of a circle? For example, we may need to know the diameter of a clock to see if it will
fit in a spot on the wall. Think of some other times knowing information about a circle will be
useful.
Investigation:
Part 1: Inscribed Angles
Define the following terms:
Inscribed Angle:
Central Angle:
Intercepted Arc:
Measure of an Inscribed Angle:
A
C
D
1. Use the figure on the left to determine the relationship between
the measure of the inscribed angle ADB and the central angle
ACB. (To determine the angle measurement, extend the lengths of
the segments and then use the protractor.)
m <ABD =
B m < ACB =
2. What is the relationship between the inscribed angle and the central angle?
3. What is the measure of arc AC? How do you know?
4. What, then, is the relationship between an inscribed angle and it’s intercepted arc?
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5. Do you think it’s true every time? Use the circle on the
right and create your own example like the one previous. Is
what you found for the first circle true for the second?
Inscribed Angle Congruency:
6. Define the inscribed angles in the circle to the left.
A
7. What do the two angles have in common?
C
B
D
8. Find the measure of the angles by extending the
segments that create the angles and using your protractor.
m < ACB =
m < BDA =
9. What is the relationship between angle ACB and angle BDA?
10. Does it hold true for every circle? Make another
example using the circle on the right and see if the same
holds true.
MM2G3b,d
11. Define the following terms:
Inscribed:
Circumscribed:
Inscribed Right Triangle:
12. If a right triangle is ________________________ in a circle, then the _________________
is a _____________________ of the circle.
13. Conversely, if one side of a(n) _____________________
B
triangle is a __________________ of the circle, then the
triangle is a ________________ triangle and the angle
opposite the diameter is the right angle.
A
C
14. Given an inscribed triangle with side lengths 12, 16, and
20 where the side 20 goes through the center of the circle,
determine if the triangle is a right triangle. If the triangle is a
right triangle, find the area of the circle. (Acircle = π r2)
Inscribed Quadrilateral:
15. A quadrilateral can be inscribed in a ________________________ if
and only if its ________________ angles are supplementary.
B
A
93
16. In the figure on the right, is the quadrilateral inscribed in the circle?
How do you know?
87
D
C
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Part 2: Properties of Chords
1. Construct two congruent chords in Circle P below. Label the chords AB and CD.
P
2. Construct the radii AP, BP, CP, and DP.
3. Measure the central angles APB and CPD .
m < APB =
m < CPD =
4. What can you conclude about congruent chords?
5. Draw a chord that is not a diameter in Circle P below. Label it AB.
P
6. Construct a line perpendicular to AB through P. Label the point of intersection of this line
7. Find the lengths of AC and BC.
8. What can you conclude?
MM2G3b,d
9. Choose a point S on Circle P below and construct four chords from the point.
P
10. Measure the length of the 4 chords and record it below:
Chord 1
Chord 2
Chord 3
Chord 4
Length
11. How are the lengths of the chords related to their distances from the center of the circle?
Conclusions:
Choose the best word to complete each definition:
1. The measure of an inscribed angle is ( one half , two times , equal to ) the measure of the
intercepted arc.
2. The measure of a central angle is ( one half , two times , equal to ) the measure of the
intercepted arc.
Write the two formulas you have discovered in terms of the intercepted arc:
Measure of a Central Angle =
Measure of an Inscribed Angle =
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In Class Problems:
1. Identify the missing measure represented by the numbers below:
Draw pictures to determine the missing lengths in the circles below:
11. A chord is 32 inches long and is 12 inches from the center of the circle. Find the Radius.
12. Find the length of a chord that is 10 feet from the center of a circle with a radius of 26 feet.
Closure:
Fill in the blank to complete each definition:
1. Inscribed angles that intercept the same arc or congruent arcs are _____________.
2. If a right triangle is inscribed in a circle, then the hypotenuse is a _____________.
3. A quadrilateral can be inscribed in a circle if and only if it’s ________________ angles
are _________________.
MM2G3b,d
Homework:
1. Find mABD and mC.
B
36°
C
A
D
2. Given CA  TA and CA =75, find the measure of  A .
A
C
T
3. Given that DAB and DCB are right angles and mCAB = 210, what is the measure of
DBC?
A
D
B
C
4. Find the values of x and y so that a circle can be circumscribed about ABCD?
C
4x - 11°
B
5y°
2x + 5°
A
80°
D
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5. Refer to the diagram. If P is the center of the circle, what is m AB ?
B
A
50º
P
C
D
6. On Survivor, two chords are stretched across a circular pit to fit the point in the sand where a
contestant needs to dig to find a clue. Once the chords are stretched, they are sectioned into 4
different expressions. The contestant must find the value x which solves each expression in order
to unlock the box under the sand. Your job is to help the survivor find the value of x.
x
(x + 5)
6
12
7. Find the missing arc measure of HS.
H
84°
92°
x°
S