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ISEE Upper Level Quantitative : Sectors

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All ISEE Upper Level Quantitative Resources

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Example Question #2 : How To Find The Length Of An Arc

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. $\angle B \cong \angle C$

Which is the greater quantity?

(a) $m \widehat{ABC}$

(b) $m \widehat{CAB}$

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

To compare $m \widehat{ABC}$ and $m \widehat{CAB}$ , we note that

$m \widehat{ABC} = m \widehat{AB} + m \widehat{BC}$

and

$m \widehat{CAB} = m \widehat{AB} + m \widehat{AC}$

We need to be able to compare $m \widehat{BC}$ and $m \widehat{AC}$ . If we know which of the intercepting angles is the greater, then we know which of the arcs is greater. The intercepting angles are $\angle A , \angle B$ , respectively. However, we are not given this relationship.

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Example Question #3 : How To Find The Length Of An Arc

A giant clock has a minute hand that is six feet long. The time is now 3:50 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

Possible Answers:

$138 \pi \textrm{ in}$

$1,104 \pi \textrm{ in}$

The correct answer is not among these choices.

$552 \pi \textrm{ in}$

$276 \pi \textrm{ in}$

Correct answer:

$552 \pi \textrm{ in}$

Explanation:

Every hour, the tip of the minute hand travels the circumference of a circle with radius six feet, which is

$C = 2 \pi r= 2 \pi \cdot 6 = 12 \pi$ feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus $\frac{50}{60 } = \frac{5}{6}$ of a fourth, so its tip has traveled this circumference $3\frac{5}{6}$ times.

This is

$3\frac{5}{6} \times 12 \pi = \frac{23}{6} \times 12 \pi = 46 \pi$ feet. This is

$46 \pi \times 12 = 552 \pi$ inches.

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

A giant clock has a minute hand seven feet long. Which is the greater quantity?

(A) The distance traveled by the tip of the minute hand between 1:30 PM and 2:00 PM

(B) The circumference of a circle seven feet in diameter

Possible Answers:

It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

(A) and (B) are equal

Correct answer:

(A) and (B) are equal

Explanation:

The tip of a minute hand travels a circle whose radius is equal to the length of that minute hand, which, in this question, is seven feet long. The circumference of this circle is $2 \pi$ times the radius, or $2 \pi \times 7 = 14\pi$ feet; over the course of thrity minutes (or one-half of an hour) the tip of the minute hand covers half this distance, or $\frac{1}{2} \times 14 \pi = 7 \pi$ feet.

The circumference of a circle seven feet in diameter is $\pi$ times this diameter, or $7 \pi$ feet.

The quantities are equal.

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Example Question #5 : How To Find The Length Of An Arc

A giant clock has a minute hand four and one-half feet in length. Since noon, the tip of the minute hand has traveled $110 \pi$ feet. Which of the following is true of the time right now?

Possible Answers:

The time is between 12:00 midnight and 12:30 AM.

The time is between 12:30 AM and 1:00 AM.

The time is between 1:00 AM and 1:30 AM.

The time is between 11:30 PM and 12:00 midnight.

The time is between 11:00 PM and 11:30 PM.

Correct answer:

The time is between 12:00 midnight and 12:30 AM.

Explanation:

Every hour, the tip of the minute hand travels the circumference of a circle, which here is

$C = 2 \pi r = 2 \pi \times 4 \frac{1}{2} = 9 \pi$ feet.

The minute hand has traveled $110 \pi$ feet since noon, so it has traveled the circumference of the circle

$\frac{110 \pi}{9 \pi} = \frac{110 }{9 } =12 \frac{2}{9}$ times.

Since $12 \leq 12 \frac{2}{9} \leq 12 \frac{1}{2}$ , between 12 and $12 \frac{1}{2}$ hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.

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

Acute triangle $\bigtriangleup ABC$ is inscribed in a circle. Which is the greater quantity?

(a) $m \overarc {ABC} - m \overarc {ACB}$

(b) $m \overarc {A B} - m \overarc {A C}$

Possible Answers:

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

Examine the figure below, which shows $\bigtriangleup ABC$ inscribed in a circle.

Inscribed angle

By the Arc Addition Principle,

$m \overarc {ABC} = m \overarc {AB}+ m \overarc {BC}$

and

$m \overarc {ACB} = m \overarc {AC}+ m \overarc {BC}$

Consequently,

$m \overarc {ABC} - m \overarc {ACB}$

$= (m \overarc {AB}+ m \overarc {BC} ) - ( m \overarc {AC}+ m \overarc {BC})$

$= m \overarc {AB}+ m \overarc {BC} - m \overarc {AC}- m \overarc {BC}$

$= m \overarc {A B} - m \overarc {A C}$

The two quantities are equal.

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

Which is the greater quantity?

(a) The degree measure of a 10-inch-long arc on a circle with radius 8 inches.

(b) The degree measure of a 12-inch-long arc on a circle with radius 10 inches.

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

(a) A circle with radius 8 inches has crircumference $C = 2\pi r = 2\pi \cdot 8 = 16\pi$ inches. An arc 10 inches long is $\frac{10}{16\pi } = \frac{5}{8\pi }$ of that circle. $\frac{5}{8\pi } \times 360 ^{\circ } = \frac{225}{\pi } ^{\circ }$ , the degree measure of this arc.

(b) A circle with radius 10 inches has crircumference $C = 2\pi r = 2\pi \cdot 10 = 20\pi$ inches. An arc 12 inches long is $\frac{12}{20\pi }=\frac{3}{5\pi }$ of that circle. $\frac{3}{5\pi } \times 360 ^{\circ } = \frac{216}{\pi } ^{\circ }$ , the degree measure of this arc.

(a) is the greater quantity.

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

Circle

Note: figure NOT drawn to scale

Refer to the above figure. Which is the greater quantity?

(a) $m \widehat{ACB}$

(b) $180 ^{\circ }$

Possible Answers:

(a) and (b) are equal

(It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

Since

$7^{2} + 12 ^{2} = 49 + 144 = 169 = 13 ^{2}$ ,

the triangle is a right triangle with right angle $\angle ACB$ .

$\angle ACB$ is an inscribed angle on the circle, so the arc it intercepts is a semicricle. Therefore, $\widehat{ACB}$ is also a semicircle, and it measures $180 ^{\circ }$ .

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Example Question #11 : Sectors

Circle

Note: Figure NOT drawn to scale

Refer to the above figure. Which is the greater quantity?

(a)

(b) 90

Possible Answers:

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

The measure of an arc intercepted by an inscribed angle of a circle is twice that of the angle. Therefore, $y = 2 \cdot 44 = 88 < 90$

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Example Question #4 : How To Find The Angle Of A Sector

$\odot A$ has twice the radius of $\odot B$ . Sector 1 is part of $\odot A$ ; Sector 2 is part of $\odot B$ ; the two sectors are equal in area.

Which is the greater quantity?

(a) Twice the degree measure of the central angle of Sector 1

(b) The degree measure of the central angle of Sector 2

Possible Answers:

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

$\odot A$ has twice the radius of $\odot B$ , so $\odot A$ has four times the area of $\odot B$ . This means that for a sector of $\odot A$ to have the same area as a sector of $\odot B$ , the central angle of the latter sector must be four times that of the former sector. This makes (b) greater than (a), which is only twice that of the former sector.

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Example Question #12 : Sectors

Circle

Refer to the above figure. Which is the greater quantity?

(a)

(b) 55

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

The measure of an inscribed angle of a circle is one-half that of the arc it intercepts. Therefore, $x = \frac{1}{2} \cdot 110 = 55$ .

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ISEE Upper Level Quantitative : Sectors

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #2 : How To Find The Length Of An Arc

Example Question #3 : How To Find The Length Of An Arc

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

Example Question #5 : How To Find The Length Of An Arc

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 #11 : Sectors

Example Question #4 : How To Find The Angle Of A Sector

Example Question #12 : Sectors

All ISEE Upper Level Quantitative Resources

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