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ISEE Upper Level Math : Circles

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Example Questions

Example Question #211 : Plane Geometry

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of $65^{\circ}$ . If the screen has a radius of 4 inches, what is the area of the highlighted wedge?

Possible Answers:

$14.23\pi in^2$

$2.89\pi in^2$

2.89 in^2

$9.08\pi in^2$

Correct answer:

$2.89\pi in^2$

Explanation:

To begin, let's recall our formula for area of a sector.

$A_{sector}=\frac{\theta}{360^{\circ}}\pi r^2$

Now, we have theta and r, so we just need to plug them in and simplify!

$A_{sector}=\frac{65^{\circ}}{360^{\circ}}\pi (4in)^2=2.89\pi in^2$

So our answer is $2.89\pi in^2$

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Example Question #211 : Plane Geometry

A giant clock has a minute hand six feet long. How far, in inches, did the tip move between noon and 1:20 PM?

Possible Answers:

$48 \pi \textrm{ in}$

$384 \pi \textrm{ in}$

$768 \pi \textrm{ in}$

$192 \pi \textrm{ in}$

$96\pi \textrm{ in}$

Correct answer:

$192 \pi \textrm{ in}$

Explanation:

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius 6 feet. This circumference is $2\pi r = 2 \pi \times 6 = 12\pi$ feet. One hour and twenty minutes is $1 \frac{1}{3}$ hours, so the tip of the hand moved $1 \frac{1}{3} \times 12 \pi = 16 \pi$ feet, or $16 \pi \times 12 = 192 \pi$ inches.

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Example Question #2 : Sectors

A giant clock has a minute hand three feet long. How far, in inches, did the tip move between noon and 12:20 PM?

Possible Answers:

$36\pi \textrm{ in}$

$12\pi \textrm{ in}$

$24\pi \textrm{ in}$

$18\pi \textrm{ in}$

It is impossible to tell from the information given

Correct answer:

$24\pi \textrm{ in}$

Explanation:

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius feet. This circumference is $2\pi r = 2 \pi \times 3 = 6\pi$ feet. minutes is one-third of an hour, so the tip of the minute hand moves $\frac{1}{3} \times 6\pi = 2\pi$ feet, or $2\pi \times 12 = 24\pi$ inches.

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Example Question #2 : Sectors

Inscribed

In the above figure, express in terms of .

Possible Answers:

$y = \frac{1}{2}x - \frac{3}{2}$

$y = x- \frac{1}{4}$

$y = \frac{1}{2}x - 10$

$y = x- \frac{27}{4}$

Correct answer:

$y = x- \frac{27}{4}$

Explanation:

The measure of an arc - $\overarc{AB}$ - intercepted by an inscribed angle - $\angle ACB$ - is twice the measure of that angle, so

$m \overarc{AB} = 2 \cdot m \angle ACB$

4y + 13 = 2 (2x- 7)

4y + 13 = 4x- 14

4y + 13 - 13 = 4x- 14 - 13

4y = 4x- 27

$\frac{4y}{4} = \frac{4x- 27}{4}$

$y = x- \frac{27}{4}$

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Example Question #213 : Geometry

Intercepted

In the above diagram, radius OA = 12 .

Give the length of $\overarc {AB}$ .

Possible Answers:

$\frac{48 \pi }{5}$

$\frac{288\pi }{5}$

$\frac{72 \pi }{5}$

$\frac{24 \pi }{5}$

Correct answer:

$\frac{48 \pi }{5}$

Explanation:

The circumference of a circle is $2 \pi$ multiplied by its radius , so

$C = 2 \pi \cdot OA = 2 \pi \cdot 12 = 24 \pi$ .

$\angle ACB$ , being an inscribed angle of the circle, intercepts an arc $\overarc {AB}$ with twice its measure:

$m \overarc {AB} = 2 \cdot m\angle ACB = 2 \cdot 72 ^{\circ } = 144 ^{\circ }$

The length of $\overarc {AB}$ is the circumference multiplied by $\frac{144}{360}$ :

$\frac{144}{360} \times 24 \pi = \frac{48 \pi }{5}$ .

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Example Question #1 : How To Find The Length Of An Arc

Possible Answers:

$3.01\pi in$

4.54in

1.44in

$4.54\pi in$

Correct answer:

4.54in

Explanation:

To begin, let's recall our formula for length of an arc.

$Arc=\frac{\theta}{360^{\circ}}2 \pi r$

Now, just plug in and simplify

$Arc=\frac{65^{\circ}}{360^{\circ}}2 \pi (4in)=\frac{13^{\circ}}{9^{\circ}}\pi in\approx 4.54in$

So, our answer is 4.54in

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Example Question #1 : How To Find The Angle Of A Sector

A giant clock has a minute hand four feet long. Since noon, the tip of the minute hand has traveled $10 \pi$ feet. What time is it now?

Possible Answers:

$1:30 \textrm{ PM}$

$1:15 \textrm{ PM}$

$12:45 \textrm{ PM}$

$1:20 \textrm{ PM}$

$1:45 \textrm{ PM}$

Correct answer:

$1:15 \textrm{ PM}$

Explanation:

The circumference of the path traveled by the tip of the minute hand over the course of one hour is:

$C = 2 \pi r = 2 \pi \cdot 4 = 8 \pi$ feet.

Since the tip of the minute hand has traveled $10 \pi$ feet since noon, the minute hand has made

$\frac{10 \pi }{8 \pi } = \frac{5}{4} = 1 \frac{1}{4}$ revolutions. Therefore, $1 \frac{1}{4}$ hours have elapsed since noon, making the time 1:15 PM.

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Example Question #1 : How To Find The Angle Of A Sector

Inscribed angle

Figure NOT drawn to scale

Refer to the above diagram. $\overarc {ABC}$ is a semicircle. Evaluate $m \overarc {BCA}$ given $\measuredangle B=90^\circ$ .

Possible Answers:

$282^{\circ }$

$302^{\circ }$

$292^{\circ }$

$312^{\circ }$

Correct answer:

$312^{\circ }$

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, $\angle B$ , which intercepts the semicircle $\overarc {ABC}$ , is such an angle. Consequently, $\bigtriangleup ABC$ is a right triangle, and $\angle A$ and $\angle C$ are complementary angles. Therefore,

$m \angle A + m \angle C = 90 ^{\circ }$

$y + \left ( \frac{y}{3}+2\right ) = 90$

$y + \frac{1}{3}y +2 = 90$

$\frac{4}{3}y +2 = 90$

$\frac{4}{3}y +2 - 2 = 90 - 2$

$\frac{4}{3}y = 88$

$\frac{3}{4} \cdot \frac{4}{3}y = \frac{3}{4} \cdot 88$

y = 66

$m \angle C = \left (\frac{y}{3}+ 2 \right ) ^{\circ }=\left ( \frac{66}{3}+ 2 \right )^{\circ }=( 22 + 2 )^{\circ }= 24^{\circ }$

Inscribed $\angle C$ intercepts an arc with twice its angle measure; this arc is $\overarc {AB}$ , so

$m \overarc {AB} = 2 \cdot m \angle C = 2 \cdot 24 ^{\circ } = 48 ^{\circ }$ .

The major arc corresponding to this minor arc, $\overarc {BCA}$ , has measure

$m \overarc {BCA} = 360 ^{\circ } - m \overarc {AB} = 360 ^{\circ } - 48^{\circ } = 312^{\circ }$

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Example Question #1 : How To Find The Angle Of A Sector

Inscribed angle

Note: Figure NOT drawn to scale

Refer to the above diagram. $\overarc {ABC}$ is a semicircle. Evaluate $m \overarc {AB}$ .

Possible Answers:

$69^{\circ }$

$74 ^{\circ }$

$79 ^{\circ }$

$64 ^{\circ }$

Correct answer:

$74 ^{\circ }$

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, $\angle B$ , which intercepts the semicircle $\overarc{ABC}$ , is such an angle. Consequently,

$m \angle B = 90 ^{\circ }$

4x-18 = 90

4x - 18 + 18 = 90 + 18

4x = 108

$4x \div 4 = 108 \div 4$

x = 27

$m \angle C = (x+10)^{\circ } = (27+10)^{\circ }= 37^{\circ }$

Inscribed $\angle C$ intercepts an arc with twice its angle measure; this arc is $\overarc{AB}$ , so

$m \widehat{AB} = 2 \cdot m \angle C = 2 \cdot 37 ^{\circ } = 74 ^{\circ }$ .

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Example Question #2 : How To Find The Angle Of A Sector

Intercepted

In the above diagram, radius AO = 20 .

Calculate the length of $\overarc {AB}$ .

Possible Answers:

$32 \pi$

$40 \pi$

$8 \pi$

$16 \pi$

Correct answer:

$16 \pi$

Explanation:

Inscribed $\angle ACB$ , which measures $72 ^{\circ }$ , intercepts an arc with twice its measure. That arc is $\overarc {AB}$ , which consequently has measure

$72 ^{\circ } \times 2 = 144 ^{\circ }$ .

This makes $\overarc {AB}$ an arc which comprises

$\frac{144}{360} = \frac{144 \div 72}{360 \div 72} = \frac{2}{5}$

of the circle.

The circumference of a circle is $2 \pi$ multiplied by its radius, so

$C = 2 \pi r = 2\pi \cdot 20 = 40 \pi$ .

The length of $\overarc {AB}$ is $\frac{2}{5}$ of this, or $\frac{2}{5} \cdot 40 \pi = 16 \pi$ .

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Circles

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #211 : Plane Geometry

Example Question #211 : Plane Geometry

Example Question #2 : Sectors

Example Question #2 : Sectors

Example Question #213 : Geometry

Example Question #1 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #2 : How To Find The Angle Of A Sector

All ISEE Upper Level Math Resources

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