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PSAT Math : Geometry

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Example Questions

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

Eric is riding a Ferris wheel. The Ferris wheel has 18 compartments, numbered in order clockwise. If compartment 1 is at 0 degrees and Eric enters compartment 13, what angle is he at?

Possible Answers:

260

240

180

280

300

Correct answer:

240

Explanation:

12 compartments further means 240 more degrees. 240 is the answer.

360/12 = 240 degrees

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

Circle2

In the figure above that includes Circle O, the measure of angle BAC is equal to 35 degrees, the measure of angle FBD is equal to 40 degrees, and the measure of arc AD is twice the measure of arc AB. Which of the following is the measure of angle CEF? The figure is not necessarily drawn to scale, and the red numbers are used to mark the angles, not represent angle measures.

Possible Answers:

80^o

75^o

60^o

30^o

110^o

Correct answer:

30^o

Explanation:

The measure of angle CEF is going to be equal to half of the difference between the measures two arcs that it intercepts, namely arcs AD and CD.

$\angle CEF=\frac{1}{2}(AD-CD)$

Thus, we need to find the measure of arcs AD and CD. Let's look at the information given and determine how it can help us figure out the measures of arcs AD and CD.

Angle BAC is an inscribed angle, which means that its meausre is one-half of the measure of the arc that it incercepts, which is arc BC.

$\angle BAC=\frac{1}{2}BC$

$35^o=\frac{1}{2}BC\rightarrow BC=70^o$

Thus, since angle BAC is 35 degrees, the measure of arc BC must be 70 degrees.

We can use a similar strategy to find the measure of arc CD, which is the arc intercepted by the inscribed angle FBD.

$40^o=\frac{1}{2}CD\rightarrow CD=80^o$

Because angle FBD has a measure of 40 degrees, the measure of arc CD must be 80 degrees.

We have the measures of arcs BC and CD. But we still need the measure of arc AD. We can use the last piece of information given, along with our knowledge about the sum of the arcs of a circle, to determine the measure of arc AD.

We are told that the measure of arc AD is twice the measure of arc AB. We also know that the sum of the measures of arcs AD, AB, CD, and BC must be 360 degrees, because there are 360 degrees in a full circle.

AD+AB+CD+BC=360^o

AD+AB+80^o+70^o=360^o

AD+AB=210^o

Because AD = 2AB, we can substitute 2AB for AD.

2AB+AB=210^o

3AB=210^o

AB=70^o

This means the measure of arc AB is 70 degrees, and the measure of arc AD is 2(70) = 140 degrees.

Now, we have all the information we need to find the measure of angle CEF, which is equal to half the difference between the measure of arcs AD and CD.

$\angle CEF=\frac{1}{2}(AD-CD)$

$\angle CEF=\frac{1}{2}(140^o-80^o)=\frac{1}{2}(60^o)$

$\angle CEF=30^o$

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

The length of an arc, , of a circle is $8\pi$ and the radius, , of the circle is . What is the measure in degrees of the central angle, $\theta$ , formed by the arc ?

Possible Answers:

90^o

60^o

120^o

45^o

30^o

Correct answer:

90^o

Explanation:

The circumference of the circle is $2\pi r$ .

$2\pi(16)=32\pi$

The length of the arc S is $8\pi$ .

A ratio can be established:

$\frac{S}{\text{circumference}}=\frac{\theta}{360^o}$

$\frac{8\pi}{32\pi}=\frac{\theta}{360^o}$

Solving for $\theta$ yields 90^o.

Note: This makes sense. Since the arc S was one-fourth the circumference of the circle, the central angle formed by arc S should be one-fourth the total degrees of a circle.

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Example Question #3 : How To Find The Angle Of A Sector

Sector

Note: Figure NOT drawn to scale.

In the above circle, x = 72 . Give the ratio of the area of the white sector to that of the gray sector.

Possible Answers:

9:2

6:1

4:1

5:1

7:2

Correct answer:

4:1

Explanation:

A $72^{\circ}$ sector is $\frac{72^{\circ }}{360^{\circ}}= \frac{1}{5}$ of the circle. The white sector is therefore $\frac{4}{5}$ of the circle, and the ratio of their areas is

$\frac{4}{5} : \frac{1}{5}$ ,

which simplifies to

4:1 .

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Example Question #331 : Geometry

Sector

Note: Figure NOT drawn to scale.

Refer to the above figure. The ratio of the area of the white sector to that of the gray sector is 5 to 1. Evaluate .

Possible Answers:

$72^{\circ }$

$54^{\circ }$

$66^{\circ }$

$60^{\circ }$

$80^{\circ }$

Correct answer:

$60^{\circ }$

Explanation:

The ratio of the areas is 5 to 1, so the white sector is one sixth of the circle. This means that the central angle of the white sector is one sixth of $\frac{1}{6} \cdot 360^{\circ} = 60^{\circ }$ .

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

Sector

Note: Figure NOT drawn to scale.

The area of the gray sector in the above circle is $6 \pi$ . The area of the white sector is $42 \pi$ . Evaluate .

Possible Answers:

$66\frac{2}{3} ^{\circ }$

$60^{\circ }$

$80^{\circ }$

$51 \frac{3}{7} ^{\circ }$

$45^{\circ }$

Correct answer:

$45^{\circ }$

Explanation:

The total area of the circle is the sum of the areas of the white and gray sectors, or

$6 \pi + 42 \pi = 48 \pi$

The gray sector takes up

$\frac{6 \pi}{48 \pi} = \frac{1}{8}$

of the circle, so the degree measure of the gray sector is equal to

$\frac{1}{8} \times 360 ^{\circ} = 45 ^{\circ}$

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Example Question #14 : Circles

Sector

Note: Figure NOT drawn to scale.

In the above circle, the length of arc $\widehat{MN}$ is $8 \pi$ , and the length of arc $\widehat{MYN}$ is $42 \pi$ . Evaluate .

Possible Answers:

$66\frac{2}{3}^{ \circ}$

$57 \frac{3}{5}^{\circ}$

$60 ^{\circ}$

$68\frac{4}{7} ^{\circ }$

$72^{\circ }$

Correct answer:

$57 \frac{3}{5}^{\circ}$

Explanation:

The circumference of the circle is the sum of the lengths of the arcs $\widehat{MN}$ and $\widehat{MYN}$ , which is

$8 \pi + 42 \pi = 50 \pi$

$\widehat{MN}$ is therefore

$\frac{8 \pi}{50 \pi } = \frac{4}{25}$

of the circle, and its degree measure is

$\frac{4}{25} \cdot 360 ^{\circ} = 57 \frac{3}{5}^{\circ}$

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

Circle

In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?

Possible Answers:

100

cannot be determined

Correct answer:

Explanation:

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

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

What is the angle of a sector of area in^2 on a circle having a radius of $15\:in$ ?

Possible Answers:

$72.00^{\circ}$

$15.22^{\circ}$

$22.92^{\circ}$

$0.06^{\circ}$

$3.00^{\circ}$

Correct answer:

$22.92^{\circ}$

Explanation:

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total 360 degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

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Example Question #21 : Circles

What is the angle of a sector that has an arc length of 13.5 on a circle of diameter ?

Possible Answers:

$194.14^{\circ}$

$10.74^{\circ}$

$128.92^{\circ}$

$35.81^{\circ}$

$14.24^{\circ}$

Correct answer:

$128.92^{\circ}$

Explanation:

The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:

$C=\pi d$

For your data, this means,

$C=12\pi$

Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:

$\frac{13.5}{12\pi}*360=128.9155039044336$

Rounded to the nearest hundredth, this is $128.92^{\circ}$ .

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PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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 #3 : How To Find The Angle Of A Sector

Example Question #331 : Geometry

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

Example Question #14 : Circles

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

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

Example Question #21 : Circles

All PSAT Math Resources

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