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12. Circles
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Geometry

12.02 Measures of arcs

Lesson
Questions
Adaptive
Worksheet
What do you remember?
1

Given that circle A is drawn to scale, identify:

a

An inscribed angle.

b

A central angle.

c

A radius.

d

A diameter.

A circle with point A as center and points B to F on the circle. points F and D creates a straigt line passing through the center. points BAC creates an acute angle, same with points DAE.points BC and DE creates a line segment not passing through the center.
2

Consider the given circle where \overline{JM} and \overline{LN} are diameters. Identify each arc as a major arc, minor arc, or semicircle. Then find its measure.

a

m\overset{\large\frown}{JK}

b

m\overset{\large\frown}{LMN}

c

m\overset{\large\frown}{KL}

d

m\overset{\large\frown}{JLN}

e

m\overset{\large\frown}{JNM}

f

m\overset{\large\frown}{ML}

g

m\overset{\large\frown}{KLM}

Circle P with diameters J M and L N. A radius P K is drawn between radius P J, and P L. Central angle K P L has a measure of 18 degrees, angle J P N has a measure of 47 degrees.
3

Describe the difference between arc measure and arc length.

4

Consider \angle ABC inscribed in circle O.

If the measure of \angle ABC is x \degree, what do we know about the measure of its intercepted arc \overset{\large\frown}{AC}?

Circle with center O. Points A, B, and C lie on the circle. Point B is between points A and C. A chord is drawn from points A to B. Another chord is drawn from points B to C. The angle the chords make at point B is labeled x degrees. The major arc between points A and C, not passing through point B, is highlighted.
5

Consider the given figure:

a

Use the Arc Addition Postulate to write an expression that represents m\overset{\large\frown}{AD}.

b

Find m\overset{\large\frown}{AD}.

Circle L. Moving clockwise, radius A L, B L, C L, D L, and E L are drawn. Central angle A L E has a measure of 105 degrees, angle E L D has a measure of 15 degrees, angle D L C has a measure of 50 degrees, and angle C L B has a measure of 65 degrees.
Let's practice
6

Consider the given figure.

a

Solve for m\angle UPT.

b

Solve for m\overset{\large\frown}{US}.

Circle P. Radii P Q, P U, P T, P S, and P R are drawn, placed in a counter clockwise direction. Minor arc Q U has a measure of 86 degrees, minor arc S R has a measure of 60 degrees and, minor arc Q R has a measure of 154 degrees.
7

Consider each figure:

a

Solve for m\angle CDB.

Circle A with diameter C D, and point B on the circle. A segment is drawn from D to B. Minor arc C B has a measure of 64 degrees.
b

Solve for m\angle SQT.

Circle Q with diameter RT, and point S on the circle. A segment is drawin from the center to S.Arc RS measures 70 degrees.
c

Solve for m\angle PSR.

Cricle S with points P, Q and R on the circle. Segments are drawn from the center to these 3 points. Segments PS and QS forms a right angle. Arc QR measures 57 degrees.
d

Solve for \angle MPN.

Chords M P, P N, and N O drawn on a circle. Minor arc M N has a measure of 70 degrees. Angle P N O has a measure of 56 degrees.
e

Solve for m\angle HKJ.

Circle K with diameter GJ drawn horizontally. Points H and L are on the circle. A segment is drawn from the center K to point H. Arc GH measures 122 degrees.
f

Solve for m\angle XFD.

A circle with diameter F D and points X and E on the circle. Segments are drawn from F to E and from E to X. E X intersects F D. Minor arc X D has a measure of 114 degrees.
8

Consider each figure:

a

Solve for m\overset{\large\frown}{CB}.

Circle A with diameter CD. Point B is on the circle. A segment is drawn from D to B. Diameter CD and segment DB creates an inscribed anle measuring 29 degrees.
b

Solve for m\overset{\large\frown}{RS}.

Circle Q with diameter R T, and point S on the circle. A segment is drawn from Q to S. Central angle S Q T has a measure of 110 degrees.
c

Solve for m\overset{\large\frown}{PRQ}.

A circle with points P, Q, and R on it. Minor arc P Q is intercepted by a right central angle. Minor arc Q R is intercepted by a central angle with a measure of 53 degrees
d

Solve for m\overset{\large\frown}{PO}.

Chords MP, PN, and NO drawn on a circle. Minor arc MN has a measure of 70 degrees. Angle PNO has a measure of 56 degrees.
e

Solve for m\overset{\large\frown}{GH}.

Circle K with diameter G J, and points H and L on the circle. A segment is drawn from K to H. Central angle H K G has a measure of 122 degrees.
f

Solve for m\overset{\large\frown}{DE}.

A circle with diameter FD. Points X and E are on the circle, and a segment is drawn from these points. Segment XE intersects the diameter. Segments are also drawn from F to X, F to E and E to D. The angle formed by the diameter and segment FE measures 53 degrees.
9

Let m\angle RPQ=(8x+16) \degree and \\m\angle SPT=(12x-32)\degree.

a

Solve for x.

b

Solve for m\angle SPT.

Circle P with radii P R, P S, P T, and P Q drawn consecutively in clockwise direction. Minor Arc S T and minor arc R Q are marked congruent.
10

Solve for x.

Chords G F and G E drawn on a circle. Angle F G E has a measure of 31 x plus 3 degrees. Major arc F E has a measure of 192 degrees.
11

Consider the diagram, assuming \angle ACD \cong \angle ECB:

a

Solve for m\overset{\large\frown}{AD}.

b

Solve for m\overset{\large\frown}{DB}.

Circle C with chord AD and chord EB are opposite each other forming triangle ACD and triangle ECB with radii CA, CE, CD and CB. Chord AD and Chord EB both have a length of 5. Arc AD measures 18x-40 degrees. Arc EB measures 12x+10 degrees. Arc AE meaures 40 degrees.
12

Given:

  • \overline{GE} is a diameter of the circle

  • m\overset{\large\frown}{GF}= (9x+18)\degree
  • m\angle E = 34 \degree
a

Solve for x.

b

Solve for m\overset{\large\frown}{FE}.

Triangle F G E inscribed in a circle. Side G E is the diameter of the circle.
13

Consider the right triangle \triangle ABC inscribed in the circle.

What do we know about the hypotenuse of \triangle ABC?

A cricle with an inscribed right triangle ABC. The hypotenuse is exactly at the center of the circle.
14

A sector of a circle has an arc length of 10\text{ cm} and a central angle of 72\degree. Find the radius of the circle.

15

A circle has a radius of 15\text{ cm}. The length of an arc is 12\text{ cm}. Find the measure of the arc in degrees.

16

A sector of a circle has an area of 28 \text{ cm}^{2}. The radius of the circle is 7\text{ cm}. Find the measure of the central angle of the sector.

17

The area of a sector is 50\text{ cm}^2 and the central angle is 60\degree. Find the radius of the circle.

Let's extend our thinking
18

Determine the relationship between the inscribed angles in the circle.

Justify your answer.

Seven inscribed angles intercepting the same arc. Moving clockwise, the angles are labeled 1, 2, 3, 4, 5, 6, and 7.
19

The graph shows the results of a survey in which students were asked what their favorite snack is.

a

What would be the arc measures associated with the burger and nachos categories?

b

Describe the kinds of arcs associated with the first category and the last category.

c

Are there any congruent arcs in this graph? Explain.

A circle graph is divided into 6 sectors. Each sectors contains a percentage. The data is as follows: Tacos, 4%; Nachos, 4%; Sandwich, 17%; Ice Cream, 25%; Burger, 36%; and Pizza, 14%.
20

The table shows the results of a survey in which students were asked where they went on vacation last year.

a

If you were to construct a circle graph of this information, what would be the arc measures associated with the beach and roadtrip categories?

b

Describe the kind of arcs associated with the beach and roadtrip categories.

c

Are there any congruent arcs in this graph? Explain.

Type of vacationNumber of friends
\text{Beach}20\%
\text{Camping}20\%
\text{Roadtrip}50\%
\text{Snow}10\%
21

String art can be made by weaving string through various geometric patterns. This particular design consists of 32 equally spaced points, where the string has been woven to form angles that reach the points 12 positions away from its vertex on either side.

a

Find the measure of an angle formed by two segments sharing a common endpoint on the circle.

b

If the angles were modified so that each endpoint is 8 positions from the vertex of its angle, find the measure of each.

c

If the angles were modified so that each endpoint is n positions from the vertex of its angle, find the measure of each.

A circle with 32 evenly spaced points on it. Chords are drawn to form angles that reach the points 12 positions away from its vertex on either side.
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Outcomes

G.PC.3

The student will solve problems, including those in context, by applying properties of circles.

G.PC.3b

Solve for arc measures and angles in a circle formed by central angles.

G.PC.3c

Solve for arc measures and angles in a circle involving inscribed angles.

G.PC.3f

Apply arc length or sector area to solve for an unknown measurement of the circle including the radius, diameter, arc measure, central angle, arc length, or sector area.

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