ISEE Upper Level Quantitative : Geometry

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

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Example Questions

Example Question #2 : 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:

The correct answer is not among these choices.

Correct answer:

Explanation:

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

 feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus  of a fourth, so its tip has traveled this circumference  times. 

This is

 feet. This is 

 inches.

Example Question #3 : 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:

(A) is greater

(B) is greater

(A) and (B) are equal

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

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  times the radius, or  feet; over the course of thrity minutes (or one-half of an hour) the tip of the minute hand covers half this distance, or  feet.

 

The circumference of a circle seven feet in diameter is  times this diameter, or  feet.

The quantities are equal.

Example Question #11 : Sectors

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

Possible Answers:

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

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 12:00 midnight and 12:30 AM.

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

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 

 feet.

The minute hand has traveled  feet since noon, so it has traveled the circumference of the circle

 times.

Since , between 12 and  hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.

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

Acute triangle  is inscribed in a circle. Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

Examine the figure below, which shows  inscribed in a circle.

Inscribed angle

By the Arc Addition Principle,

and

Consequently,

The two quantities are equal. 

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.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

(a) A circle with radius 8 inches has crircumference  inches. An arc 10 inches long is  of that circle. , the degree measure of this arc.

(b) A circle with radius 10 inches has crircumference  inches. An arc 12 inches long is  of that circle. , the degree measure of this arc.

(a) is the greater quantity.

 

Example Question #181 : Geometry

Circle

Note: figure NOT drawn to scale

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

(a) 

(b) 

Possible Answers:

(a) is greater

(a) and (b) are equal

(It is impossible to tell from the information given

(b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

Since 

,

the triangle is a right triangle with right angle .

 is an inscribed angle on the circle, so the arc it intercepts is a semicricle. Therefore,  is also a semicircle, and it measures .

Example Question #3 : 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) 

(b) 90

Possible Answers:

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

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, 

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

 has twice the radius of . Sector 1 is part of ; Sector 2 is part of ; 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) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

(b) is greater

Explanation:

 has twice the radius of , so  has four times the area of . This means that for a sector of  to have the same area as a sector of , 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.

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

Circle

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

(a) 

(b) 55

Possible Answers:

(a) is greater

(b) 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, .

Example Question #184 : Plane Geometry

Circlesectorgeneral81

The arc-length for the shaded sector is .  What is the value of , rounded to the nearest hundredth?

Possible Answers:

˚

˚

˚

˚

˚

Correct answer:

˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total  degrees of the circle.  The proportion of  to  is the same as  to the total circumference of the circle.

The circumference of a circle is found by:

For our data, this means:

Now we can solve for  using the proportions:

Cross multiply:

Divide both sides by :

Therefore,  is ˚.

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