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Name_______________________________ Circles Test Review4.26.17

Circles – Summary of Key Formulas and Theorems

Below is a summary of some of the key theorems from Chapter 8. This list is not necessarily a comprehensive list of all the theorems we’ve seen in Chapter 8.

Key Terms: (listed on p699)arc, arc length, central angle, chord, diameter, minor arc, major arc, arc measure, inscribed angle, circumscribed circle, inscribed circle, cyclic quadrilateral, secant, secant angle, tangent

Area of a circle: A=pr 2 Circumference of a circle: C=2 pr

Sector Area: A=pr 2· q

360 Arc Length: L=2 pr· q

360

Circle Angle-Arc Measure Relationships

Central angle – an angle formed by radii of a circle.

Inscribed angle – an angle formed by connecting two points on the circumference of a circle to another point on the circumference.

Circle Properties/Theorems:1. Central angle theorem

A central angle has the same measure as its intercepted arc.

2. Inscribed angle theoremAn inscribed angle is half the measure of its intercepted arc.

b. An angle inscribed in a semicircle is a right angle.

O is the central angle with intercepted arc AB

B

AO

C

BA

C is an inscribed angle with intercepted arc AB

B

C

AO

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c. Inscribed angles that have the same intercepted arc are equal.

∠ A=∠B Both have intercepted arc CD

3. Two tangents drawn to a circle will be equal in length. (8.12)

4. In inscribed quadrilaterals (cyclic quadrilaterals) opposite angles are supplementary. (8.11)

5. Congruent chords have congruent arcs and congruent arcs have congruent chords. (8.10)

6. A line through the center of a circle and perpendicular to a chord bisects the chord.

AND A line through the center of a circle that bisects a chord is perpendicular to the chord. (8.10)

7. A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. (8.12)

C = 90°

DC

B

A

= D

Both have the same intercepted arc AB

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Equation of circle: (x – h)2 + (y – k)2 = r2

Directions: Do all work on a separate piece of paper.

1. Find the specified missing measures in circle A. Note that the 60 degree mark in the picture means that the measure of arc EC is 60 degrees.

a. mCAE = _________

b. mADE = _________

c. mDEA = _________

2. Find the specified missing measures in circle A.

a. mDAF = _________

b. Area of sector DAF = ___________

3. a. Put this circle equation into the form that allows you to identify the center and radius:

x2 – 6x + y2 + 4y = 12

b. Find the coordinates of the points at the top, bottom, left, and right of this circle.

4. On a circle of radius 10, there is an arc whose length is 4 π . What is the degree measure of this arc and its central angle?

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5. Shown on the grid below are circles centered at A and B, and a line t that is tangent to both circles.

a. Compare the slopes of line AB and line t. How are they related?

b. Write an x-y equation for each circle and for line t.

6. The last diagram shows circles centered at C and D, two lines tangent to the circles, and some radii.

a. Prove ΔCQX ΔCRX.

b. Prove QT = RS.

7. Given: BC is tangent to circle A at point B, and tangent to circle D at point C. E is the midpoint of BC .

Prove: Circle A is congruent to Circle D.

Proof:

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8. Find the missing measures and lengths.KJ is a tangent segment, and goes through points H and J.The radius of circle F is 7. The radius of circle J is 18. m∠KFG=75 ° .

9. Find the shaded region of circle C to the right. Note that the circle has radius 3 inches.

10. Given: Circle P tangent to circle O and ∠BOA=30 ° . Find the measure of arc BF. Write and justify each step you take to find the answer.

11.

FJ=¿¿

JK=¿¿

m∠KJF=¿¿

mHG=¿ ¿

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More practice applying the theorems:2. 3.

If you would like extra practice complete the following problems (these are not required) p699 #1-6

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Web viewarc, arc length, central angle, chord, diameter, minor arc, major arc, arc measure, inscribed angle, circumscribed circle, inscribed circle, cyclic quadrilateral, secant, secant