Geometry Quiz: Circle Relationships Angles Radii And Chords

9 questions · exam conditions

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Circle Relationships Angles Radii And ChordsQuestion 1 of 9

In circle $O$ , tangent $PT$ touches the circle at point $T$ , and secant $PAB$ passes through the circle with $A$ and $B$ on the circle. If $PT = 12$ and $PA = 8$ , what is the length of $PB$ ?

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Geometry Quiz

Geometry Quiz: Circle Relationships Angles Radii And Chords

Practice Circle Relationships Angles Radii And Chords in Geometry with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Circle Relationships Angles Radii And Chords, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

In circle $O$ , tangent $PT$ touches the circle at point $T$ , and secant $PAB$ passes through the circle with $A$ and $B$ on the circle. If $PT = 12$ and $PA = 8$ , what is the length of $PB$ ?

16
18 (correct answer)
20
24

Explanation: By the tangent-secant theorem, when a tangent and secant are drawn from the same external point,

PT^2 = PA \cdot PB

. Substituting the known values:

12^2 = 8 \cdot PB

, so

144 = 8 \cdot PB

, giving

PB = 18

. Choice A (16) might result from incorrectly using

PT \cdot PA = PB

. Choice C (20) could come from using

PT + PA = PB

. Choice D (24) might result from using

2 \cdot PT = PB

Question 2

Refer to the figure below. In circle $O$ , $\overline{AB}$ is a diameter, and $C$ is a point on the circle such that $\overline{AC} = 8$ and $\overline{BC} = 15$ . What is the radius of circle $O$ ?

$7.5$
$8.5$ (correct answer)
$11.5$
$17$

Explanation: Since

\overline{AB}

is a diameter, the inscribed angle

\angle ACB = 90°

. Then

AB = \sqrt{8^2 + 15^2} = \sqrt{289} = 17

, so the radius is

17/2 = 8.5

. Distractor A is half of 15 (mistaking the longer leg for the diameter). C averages the legs. D is the diameter, not the radius.

Question 3

Refer to the figure below. Two tangent lines from external point $P$ touch circle $O$ at points $A$ and $B$ . If $\angle APB = 46°$ , what is the measure of the minor arc $AB$ ?

$46°$
$67°$
$134°$ (correct answer)
$314°$

Explanation: Radii

\overline{OA}

and

\overline{OB}

are perpendicular to the tangents, so quadrilateral

OAPB

has two right angles. The remaining two angles sum to

180°

, giving central angle

\angle AOB = 180° - 46° = 134°

, which equals the minor arc

AB

. Distractor A copies the given angle. B is half the correct central angle. D is the major arc.

Question 4

In the figure shown, $\angle APB$ is a central angle measuring $80¬∞$ , and $\angle AQB$ is an inscribed angle that intercepts the same arc $AB$ . Point $Q$ is then moved to a new position $Q'$ on the circle such that $\angle AQ'B$ intercepts the major arc $AB$ . What is the measure of $\angle AQ'B$ ?

40¬∞
100¬∞
140¬∞ (correct answer)
280¬∞

Explanation: Initially, inscribed angle

\angle AQB

intercepts the minor arc

AB

and measures half the central angle:

\angle AQB = \frac{80¬∞}{2} = 40¬∞

. When

Q

moves to

Q'

such that

\angle AQ'B

intercepts the major arc

AB

, this inscribed angle measures half of the major arc. The major arc

AB

measures

360¬∞ - 80¬∞ = 280¬∞

, so

\angle AQ'B = \frac{280¬∞}{2} = 140¬∞

. Choice A (40¬∞) is the measure of the original inscribed angle. Choice B (100¬∞) might result from incorrectly adding

80¬∞ + 20¬∞

. Choice D (280¬∞) is the arc measure, not the inscribed angle measure.

Question 5

In the diagram, $AB$ is a diameter of circle $O$ , and $CD$ is a chord perpendicular to $AB$ at point $E$ . If $AE = 16$ , $EB = 4$ , what is the length of $CD$ ?

8
12
16 (correct answer)
20

Explanation: Since

AB

is a diameter,

AB = AE + EB = 16 + 4 = 20

, so the radius is 10. The center

O

is at the midpoint of diameter

AB

, so

AO = OB = 10

. Since

AE = 16

and

AO = 10

, point

E

6

units from the center (

OE = |AE - AO| = |16 - 10| = 6

). When a chord is perpendicular to a diameter, the diameter bisects the chord. Using the relationship for intersecting chords:

AE \cdot EB = CE \cdot ED

. Since

CD

is bisected at

E

, let

CE = ED = x

, so

x^2 = 16 \cdot 4 = 64

, giving

x = 8

. Therefore,

CD = CE + ED = 8 + 8 = 16

. Choice A (8) is half the chord length. Choice B (12) might result from using

AE - EB = 12

. Choice D (20) is the diameter length.

Question 6

An inscribed angle and a central angle both intercept the same arc of a circle. If the central angle measures $3x + 20°$ and the inscribed angle measures $2x - 5°$ , what is the value of $x$ ?

15
25
30 (correct answer)
35

Explanation: An inscribed angle measures half of the central angle that intercepts the same arc. Therefore:

2x - 5 = \frac{1}{2}(3x + 20)

. Solving:

2x - 5 = \frac{3x + 20}{2}

, so

2(2x - 5) = 3x + 20

, giving

4x - 10 = 3x + 20

, and

x = 30

. Choice A (15) might result from setting the angles equal:

2x - 5 = 3x + 20

. Choice B (25) could come from solving

2x = 3x + 20 - 5

. Choice D (35) might result from incorrectly manipulating the equation.

Question 7

Two chords $AB$ and $CD$ intersect inside circle $O$ at point $E$ . If $AE = 6$ , $EB = 4$ , and $CE = 8$ , what is the length of $ED$ ?

3 (correct answer)
2
5
12

Explanation: When two chords intersect inside a circle, the products of their segments are equal:

AE \cdot EB = CE \cdot ED

. Substituting:

6 \cdot 4 = 8 \cdot ED

, so

24 = 8 \cdot ED

, giving

ED = 3

. Choice B (2) might result from using

AE - EB = ED

. Choice C (5) could come from incorrectly using

\frac{AE + EB}{2} - 1 = ED

. Choice D (12) might result from using

CE + EB = ED

Question 8

In circle $M$ , radius $MR$ is perpendicular to chord $ST$ at point $H$ . If the radius of the circle is 13 and $MH = 5$ , what is the length of chord $ST$ ?

12
18
24 (correct answer)
26

Explanation: When a radius is perpendicular to a chord, it bisects the chord. In right triangle

MHS

(where

S

is one endpoint of the chord), we have

MS = 13

(radius),

MH = 5

, and

HS

can be found using the Pythagorean theorem:

HS^2 + MH^2 = MS^2

, so

HS^2 + 25 = 169

, giving

HS^2 = 144

and

HS = 12

. Since the radius bisects the chord,

ST = 2 \cdot HS = 2 \cdot 12 = 24

. Choice A (12) is just half the chord length. Choice B (18) might result from adding

MH + MS = 5 + 13 = 18

. Choice D (26) might come from adding

2 \cdot MS = 2 \cdot 13 = 26

Question 9

In circle $P$ , two secants are drawn from external point $R$ . Secant $RST$ has $RS = 9$ and $ST = 7$ . Secant $RUV$ has $RU = 6$ and $UV = x$ . What is the value of $x$ ?

8
12
18 (correct answer)
21

Explanation: By the secant-secant theorem, when two secants are drawn from the same external point:

RS \cdot RT = RU \cdot RV

, where

RT = RS + ST

and

RV = RU + UV

. So:

RT = 9 + 7 = 16

and

RV = 6 + x

. The equation becomes:

9 \cdot 16 = 6 \cdot (6 + x)

, so

144 = 6(6 + x) = 36 + 6x

, giving

6x = 108

and

x = 18

. Choice A (8) might result from using

RS - RU + ST = x

. Choice B (12) could come from incorrectly using

\frac{RS \cdot ST}{RU} = x

. Choice D (21) might result from using

RS + ST + RU - 1 = x