Circles 1

1

In the diagram shown below, $\overline{AC}$ is tangent to circle $O$ at $A$ and to circle $P$ at $C$, $\overline{OP}$ intersects $\overline{AC}$ at $B$, $OA = 4$, $AB = 5$, and $PC = 10$.

What is the length of $\overline{BC}$?

6.4

8

12.5

16

1

In the diagram below, quadrilateral $ABCD$ is inscribed in circle $P$.

What is $m\angle ADC$?

$70^\circ$

$72^\circ$

$108^\circ$

$110^\circ$

1

In circle $O$ shown below, diameter $\overline{AC}$ is perpendicular to $\overline{CD}$ at point $C$, and chords $\overline{AB}$, $\overline{BC}$, $\overline{AE}$, and $\overline{CE}$ are drawn.

Which statement is not always true?

$\angle ACB \cong \angle BCD$

$\angle ABC \cong \angle ACD$

$\angle BAC \cong \angle DCB$

$\angle CBA \cong \angle AEC$

1

In the diagram below, $\overline{BC}$ is the diameter of circle $A$.

Point $D$, which is unique from points $B$ and $C$, is plotted on circle $A$. Which statement must always be true?

$\triangle BCD$ is a right triangle.

$\triangle BCD$ is an isosceles triangle.

$\triangle BAD$ and $\triangle CBD$ are similar triangles.

$\triangle BAD$ and $\triangle CAD$ are congruent triangles.

2

Lines $\overline{AE}$ and $\overline{BD}$ are tangent to circles $O$ and $P$ at $A$, $E$, $B$, and $D$, as shown in the diagram below. If $AC:CE = 5:3$, and $\overline{BD} = 56$, determine and state the length of $\overline{CD}$.

1

In the diagram below, $\overline{DC}$, $\overline{AC}$, $\overline{DOB}$, $\overline{CB}$, and $\overline{AB}$ are chords of circle $O$, $\overline{FDE}$ is tangent at point $D$, and radius $\overline{AO}$ is drawn. Sam decides to apply this theorem to the diagram: “An angle inscribed in a semi-circle is a right angle.”
Which angle is Sam referring to?

$\angle AOB$

$\angle BAC$

$\angle DCB$

$\angle FDB$

2

In the diagram below of circle $O$ with diameter $\overline{BC}$ and radius $\overline{OA}$, chord $\overline{DC}$ is parallel to chord $\overline{BA}$.
If $m\angle BCD = 30^\circ$, determine and state $m\angle AOB$.

1

In circle $O$, secants $\overline{ADB}$ and $\overline{AEC}$ are drawn from external point $A$ such that points $D$, $B$, $E$, and $C$ are on circle $O$. If $\overline{AD} = 8$, $\overline{AE} = 6$, and $\overline{EC}$ is 12 more than $\overline{BD}$, the length of $\overline{BD}$ is

6

22

36

48

2

In the diagram below, tangent $\overline{DA}$ and secant $\overline{DBC}$ are drawn to circle $O$ from external point $D$, such that $\widehat{AC} \cong \widehat{BC}$.
If $m\widehat{BC} = 152^\circ$, determine and state $m\angle D$.

Circles 1

In the diagram shown below, $\overline{AC}$ is tangent to circle $O$ at $A$ and to circle $P$ at $C$, $\overline{OP}$ intersects $\overline{AC}$ at $B$, $OA = 4$, $AB = 5$, and $PC = 10$.What is the length of $\overline{BC}$?

In the diagram below, quadrilateral $ABCD$ is inscribed in circle $P$.What is $m\angle ADC$?

In circle $O$ shown below, diameter $\overline{AC}$ is perpendicular to $\overline{CD}$ at point $C$, and chords $\overline{AB}$, $\overline{BC}$, $\overline{AE}$, and $\overline{CE}$ are drawn.Which statement is not always true?

In the diagram below, $\overline{BC}$ is the diameter of circle $A$.Point $D$, which is unique from points $B$ and $C$, is plotted on circle $A$. Which statement must always be true?

Lines $\overline{AE}$ and $\overline{BD}$ are tangent to circles $O$ and $P$ at $A$, $E$, $B$, and $D$, as shown in the diagram below. If $AC:CE = 5:3$, and $\overline{BD} = 56$, determine and state the length of $\overline{CD}$.

In the diagram below of circle $O$ with diameter $\overline{BC}$ and radius $\overline{OA}$, chord $\overline{DC}$ is parallel to chord $\overline{BA}$.If $m\angle BCD = 30^\circ$, determine and state $m\angle AOB$.

In circle $O$, secants $\overline{ADB}$ and $\overline{AEC}$ are drawn from external point $A$ such that points $D$, $B$, $E$, and $C$ are on circle $O$. If $\overline{AD} = 8$, $\overline{AE} = 6$, and $\overline{EC}$ is 12 more than $\overline{BD}$, the length of $\overline{BD}$ is

In the diagram below, tangent $\overline{DA}$ and secant $\overline{DBC}$ are drawn to circle $O$ from external point $D$, such that $\widehat{AC} \cong \widehat{BC}$.If $m\widehat{BC} = 152^\circ$, determine and state $m\angle D$.

In the diagram shown below, $\overline{AC}$ is tangent to circle $O$ at $A$ and to circle $P$ at $C$, $\overline{OP}$ intersects $\overline{AC}$ at $B$, $OA = 4$, $AB = 5$, and $PC = 10$.What is the length of $\overline{BC}$?

In the diagram below, quadrilateral $ABCD$ is inscribed in circle $P$.What is $m\angle ADC$?

In circle $O$ shown below, diameter $\overline{AC}$ is perpendicular to $\overline{CD}$ at point $C$, and chords $\overline{AB}$, $\overline{BC}$, $\overline{AE}$, and $\overline{CE}$ are drawn.Which statement is not always true?

In the diagram below, $\overline{BC}$ is the diameter of circle $A$.Point $D$, which is unique from points $B$ and $C$, is plotted on circle $A$. Which statement must always be true?

Lines $\overline{AE}$ and $\overline{BD}$ are tangent to circles $O$ and $P$ at $A$, $E$, $B$, and $D$, as shown in the diagram below. If $AC:CE = 5:3$, and $\overline{BD} = 56$, determine and state the length of $\overline{CD}$.

In the diagram below of circle $O$ with diameter $\overline{BC}$ and radius $\overline{OA}$, chord $\overline{DC}$ is parallel to chord $\overline{BA}$.If $m\angle BCD = 30^\circ$, determine and state $m\angle AOB$.

In circle $O$, secants $\overline{ADB}$ and $\overline{AEC}$ are drawn from external point $A$ such that points $D$, $B$, $E$, and $C$ are on circle $O$. If $\overline{AD} = 8$, $\overline{AE} = 6$, and $\overline{EC}$ is 12 more than $\overline{BD}$, the length of $\overline{BD}$ is

In the diagram below, tangent $\overline{DA}$ and secant $\overline{DBC}$ are drawn to circle $O$ from external point $D$, such that $\widehat{AC} \cong \widehat{BC}$.If $m\widehat{BC} = 152^\circ$, determine and state $m\angle D$.

In the diagram shown below, $\overline{AC}$ is tangent to circle $O$ at $A$ and to circle $P$ at $C$, $\overline{OP}$ intersects $\overline{AC}$ at $B$, $OA = 4$, $AB = 5$, and $PC = 10$.

What is the length of $\overline{BC}$?

In the diagram below, quadrilateral $ABCD$ is inscribed in circle $P$.

What is $m\angle ADC$?

In circle $O$ shown below, diameter $\overline{AC}$ is perpendicular to $\overline{CD}$ at point $C$, and chords $\overline{AB}$, $\overline{BC}$, $\overline{AE}$, and $\overline{CE}$ are drawn.

Which statement is not always true?

In the diagram below, $\overline{BC}$ is the diameter of circle $A$.

Point $D$, which is unique from points $B$ and $C$, is plotted on circle $A$. Which statement must always be true?

In the diagram below of circle $O$ with diameter $\overline{BC}$ and radius $\overline{OA}$, chord $\overline{DC}$ is parallel to chord $\overline{BA}$.
If $m\angle BCD = 30^\circ$, determine and state $m\angle AOB$.

In the diagram below, tangent $\overline{DA}$ and secant $\overline{DBC}$ are drawn to circle $O$ from external point $D$, such that $\widehat{AC} \cong \widehat{BC}$.
If $m\widehat{BC} = 152^\circ$, determine and state $m\angle D$.

In the diagram shown below, $\overline{AC}$ is tangent to circle $O$ at $A$ and to circle $P$ at $C$, $\overline{OP}$ intersects $\overline{AC}$ at $B$, $OA = 4$, $AB = 5$, and $PC = 10$.

What is the length of $\overline{BC}$?

In the diagram below, quadrilateral $ABCD$ is inscribed in circle $P$.

What is $m\angle ADC$?

In circle $O$ shown below, diameter $\overline{AC}$ is perpendicular to $\overline{CD}$ at point $C$, and chords $\overline{AB}$, $\overline{BC}$, $\overline{AE}$, and $\overline{CE}$ are drawn.

Which statement is not always true?

In the diagram below, $\overline{BC}$ is the diameter of circle $A$.

Point $D$, which is unique from points $B$ and $C$, is plotted on circle $A$. Which statement must always be true?

In the diagram below of circle $O$ with diameter $\overline{BC}$ and radius $\overline{OA}$, chord $\overline{DC}$ is parallel to chord $\overline{BA}$.
If $m\angle BCD = 30^\circ$, determine and state $m\angle AOB$.

In the diagram below, tangent $\overline{DA}$ and secant $\overline{DBC}$ are drawn to circle $O$ from external point $D$, such that $\widehat{AC} \cong \widehat{BC}$.
If $m\widehat{BC} = 152^\circ$, determine and state $m\angle D$.