ISEE Upper Level Quantitative : Plane 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?
The correct answer is not among these choices.
Every hour, the tip of the minute hand travels the circumference of a circle with radius six feet, which is
Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus 
This is
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
(A) is greater
(B) is greater
(A) and (B) are equal
It is impossible to determine which is greater from the information given
(A) and (B) are equal
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 
The circumference of a circle seven feet in diameter is 
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 
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.
The time is between 12:00 midnight and 12:30 AM.
Every hour, the tip of the minute hand travels the circumference of a circle, which here is
The minute hand has traveled 
Since 
Example Question #6 : How To Find The Length Of An Arc
Acute triangle 
(a)
(b)
It is impossible to determine which is greater from the information given
(a) is the greater quantity
(a) and (b) are equal
(b) is the greater quantity
(a) and (b) are equal
Examine the figure below, which shows 
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.
(b) is greater.
(a) and (b) are equal.
(a) is greater.
It is impossible to tell from the information given.
(a) is greater.
(a) A circle with radius 8 inches has crircumference 
(b) A circle with radius 10 inches has crircumference 
(a) is the greater quantity.
Example Question #2 : How To Find The Angle Of A Sector
Note: figure NOT drawn to scale
Refer to the above figure. Which is the greater quantity?
(a)
(b)
(a) is greater
(It is impossible to tell from the information given
(a) and (b) are equal
(b) is greater
(a) and (b) are equal
Since
the triangle is a right triangle with right angle 
Example Question #3 : How To Find The Angle Of A Sector
Note: Figure NOT drawn to scale
Refer to the above figure. Which is the greater quantity?
(a)
(b) 90
(a) and (b) are equal
It is impossible to tell from the information given
(a) is greater
(b) is greater
(b) is greater
The measure of an arc intercepted by an inscribed angle of a circle is twice that of the angle. Therefore,
Example Question #184 : Geometry
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
(a) is greater
(a) and (b) are equal
It is impossible to tell from the information given
(b) is greater
(b) is greater
Example Question #14 : Sectors
Refer to the above figure. Which is the greater quantity?
(a)
(b) 55
(a) is greater
It is impossible to tell from the information given
(a) and (b) are equal
(b) is greater
(a) and (b) are equal
The measure of an inscribed angle of a circle is one-half that of the arc it intercepts. Therefore, 
Example Question #184 : Plane Geometry
The arc-length for the shaded sector is 
Remember that the angle for a sector or arc is found as a percentage of the total 
The circumference of a circle is found by:
For our data, this means:
Now we can solve for 
Cross multiply:
Divide both sides by 
Therefore, 
All ISEE Upper Level Quantitative Resources
