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Geometry A Common Core Curriculum

Ron Larson, Laurie Boswell

Chapter 10

Circles

Educators


Problem 1

How are chords and secants alike? How are they different?

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Problem 2

Explain how you can determine from the context whether the words radius and diameter are referring to segments or lengths.

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Problem 3

Coplanar circles that have a common center are called _______.

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Problem 4

Which segment does not belong with the other three? Explain your reasoning.

Martha R.
Numerade Educator

Problem 5

Use the diagram.
Name the circle.

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Problem 6

Use the diagram.
Name two radii.

Martha R.
Numerade Educator

Problem 7

Use the diagram.
Name two chords.

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Problem 8

Use the diagram.
Name a diameter.

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Numerade Educator

Problem 9

Use the diagram.
Name a secant.

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Problem 10

Use the diagram.
Name a tangent and a point of tangency.

Martha R.
Numerade Educator

Problem 11

Copy the diagram. Tell how many common tangents the circles have and draw them.

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Problem 12

Copy the diagram. Tell how many common tangents the circles have and draw them.

Martha R.
Numerade Educator

Problem 13

Copy the diagram. Tell how many common tangents the circles have and draw them.

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Problem 14

Copy the diagram. Tell how many common tangents the circles have and draw them.

Martha R.
Numerade Educator

Problem 15

Tell whether the common tangent is internal or external.

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Problem 16

Tell whether the common tangent is internal or external.

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Problem 17

Tell whether the common tangent is internal or external.

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Problem 18

Tell whether the common tangent is internal or external.

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Problem 19

Tell whether $\overline{\mathrm{AB}}$ is tangent to $\odot \mathrm{C}$ . Explain your reasoning.

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Problem 20

Tell whether $\overline{\mathrm{AB}}$ is tangent to $\odot \mathrm{C}$ . Explain your reasoning.

Martha R.
Numerade Educator

Problem 21

Tell whether $\overline{\mathrm{AB}}$ is tangent to $\odot \mathrm{C}$ . Explain your reasoning.

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Problem 22

Tell whether $\overline{\mathrm{AB}}$ is tangent to $\odot \mathrm{C}$ . Explain your reasoning.

Martha R.
Numerade Educator

Problem 23

Point $\mathrm{B}$ is a point of tangency. Find the radius $\mathrm{r}$ of $\odot \mathrm{C}$.

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Problem 24

Point $\mathrm{B}$ is a point of tangency. Find the radius $\mathrm{r}$ of $\odot \mathrm{C}$.

Martha R.
Numerade Educator

Problem 25

Point $\mathrm{B}$ is a point of tangency. Find the radius $\mathrm{r}$ of $\odot \mathrm{C}$.

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Problem 26

Point $\mathrm{B}$ is a point of tangency. Find the radius $\mathrm{r}$ of $\odot \mathrm{C}$.

Martha R.
Numerade Educator

Problem 27

Construct $\odot \mathrm{C}$ with the given radius and point A outside of $\odot \mathrm{C}$ . Then construct a line tangent to $\odot \mathrm{C}$ that passes through $\mathrm{A}$.
$\mathrm{r}=2 \mathrm{in.}$

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Problem 28

Construct $\odot \mathrm{C}$ with the given radius and point A outside of $\odot \mathrm{C}$ . Then construct a line tangent to $\odot \mathrm{C}$ that passes through $\mathrm{A}$.
$\mathrm{r}=4.5 \mathrm{cm}$

Martha R.
Numerade Educator

Problem 29

Points $\mathrm{B}$ and $\mathrm{D}$ are points of tangency. Find the value(s) of $\mathrm{x}$.

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Problem 30

Points $\mathrm{B}$ and $\mathrm{D}$ are points of tangency. Find the value(s) of $\mathrm{x}$.

Martha R.
Numerade Educator

Problem 31

Points $\mathrm{B}$ and $\mathrm{D}$ are points of tangency. Find the value(s) of $\mathrm{x}$.

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Problem 32

Points $\mathrm{B}$ and $\mathrm{D}$ are points of tangency. Find the value(s) of $\mathrm{x}$.

Martha R.
Numerade Educator

Problem 33

Describe and correct the error in determining whether $\overline{X Y}$ is tangent to $\odot Z$.
Because $11^{2}+6 \underline{0}^{2} \square 61^{2}, \Delta \mathrm{XYZ}$ a right triangle. So, $\overline{X Y}$ is tangent to $\odot Z$

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Problem 34

Describe and correct the error in finding the radius of$\odot$T.
$39^{2} \square 36^{2} \square 15^{2}$ So, the radius is 15

Martha R.
Numerade Educator

Problem 35

For a point outside of a circle, how many lines exist tangent to the circle that pass through the point? How many such lines exist for $\square$ a point on the circle? inside the circle? Explain your reasoning.

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Problem 36

When will two lines tangent to the same circle not intersect? Justify your answer.

Martha R.
Numerade Educator

Problem 37

Each side of quadrilateral TVWXis tangent to $\odot Y$. Find the perimeter of the quadrilateral.

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Problem 38

$\operatorname{In} \odot C, \operatorname{radii} \overline{C A}$ and $\overline{C B}$ are perpendicular. BD and AD are tangent to $\odot \mathrm{C}.$

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Problem 39

Two bike paths are tangent to an approximately circular pond. Your class is building a nature trail that begins at the intersection B of the bike paths and runs between the bike paths and over a bridge through the center P of the pond. Your classmate uses the Converse of the Angle Bisector Theorem (Theorem 6.4) to conclude that the trail must bisect the angle formed by the bike paths. Is your classmate correct? Explain your reasoning

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Problem 40

Asbicycle chain is $\square$ pulled tightly so that $\overline{M N}$ is a common tangent of the gears. Find the distance between the centers of the $\square$ gears.

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Problem 41

Explain why the diameter of a circle is the longest chord of the circle.

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Problem 42

In the figure, PA is tangent to the dime, PC is tangent to the quarter, and PB is a common internal tangent. How do you know that $\overline{\mathrm{PA}} \cong \overline{\mathrm{PB}} \cong \overline{\mathrm{PC}} ?$

Martha R.
Numerade Educator

Problem 43

In the diagram, $\overline{\mathrm{RS}}$ is a common internal tangent to $\odot A$ and $\odot B .$ Prove that $\frac{A C}{B C}=\frac{R C}{S C}$

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Problem 44

A polygon is circumscribed about a circle when every side of the polygon is tangent to the circle. In the diagram, quadrilateral ABCD is circumscribed about $\odot Q$ . Is it always truethat $A B \square C
D=A D \square$ BC? Justify your answer.

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Problem 45

Find the values of x and y. Justify your answer.

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Problem 46

Prove the External Tangent Congruence Theorem (Theorem 10.2).
Given $\overline{\mathrm{SR}}$ and $\overline{\mathrm{ST}}$ are tangent to $\odot$ P.
Prove $\overline{\mathrm{SR}} \cong \overline{\mathrm{ST}}$

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Problem 47

Use the diagram to prove each part of the biconditional in the Tangent Line to Circle Theorem (Theorem 10.1).
a. Prove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. (Hint: If you assume line m is not perpendicular to $\overline{\mathrm{QP}}$ , then the perpendicular segment from point Q to line m must intersect line mat some other point $\mathrm{R} )$
Given Line $\mathrm{m}$ is tangent to $\odot \mathrm{Q}$ at point $\mathrm{P}$ .
Prove $\mathrm{m} \perp \overline{\mathrm{QP}}$
b. Prove indirectly that if a line is perpendicular to a radius at its endpoint, then the line is tangent to the circle.
Given $\quad \mathrm{m} \perp \overline{\mathrm{QP}}$
Prove Line m is tangent to $\odot Q$ .

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Problem 48

In the diagram, $\mathrm{AB}=\mathrm{AC}=12, \mathrm{BC}=8$ and all three segments are tangent to \odotP. What is the radius of $\odot$P? Justify your answer.

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Problem 49

Find the indicated measure.
$\mathrm{m} \angle \mathrm{JKM}$

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Problem 50

Find the indicated measure.
$A B$

Martha R.
Numerade Educator