All ISEE Upper Level Quantitative Resources
Example Questions
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)
(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
(a) and (b) are equal
Examine the figure below, which shows inscribed in a circle.
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.
(a) and (b) are equal.
It is impossible to tell from the information given.
(b) is greater.
(a) is greater.
(a) is greater.
(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 #1 : 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) and (b) are equal
(It is impossible to tell from the information given
(b) is greater
(a) is greater
(a) and (b) are equal
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 #11 : Sectors
Note: Figure NOT drawn to scale
Refer to the above figure. Which is the greater quantity?
(a)
(b) 90
It is impossible to tell from the information given
(a) is greater
(b) is greater
(a) and (b) are equal
(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 #4 : 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
(a) is greater
(b) is greater
(a) and (b) are equal
It is impossible to tell from the information given
(b) is greater
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 #12 : Sectors
Refer to the above figure. Which is the greater quantity?
(a)
(b) 55
(b) is greater
(a) is greater
It is impossible to tell from the information given
(a) and (b) are equal
(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 #1 : How To Find The Angle Of A Sector
The arc-length for the shaded sector is . What is the value of , rounded to the nearest hundredth?
˚
˚
˚
˚
˚
˚
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 ˚.
Example Question #7 : How To Find The Angle Of A Sector
The arc-length for the shaded sector is . What is the value of , rounded to the nearest hundredth?
˚
˚
˚
˚
˚
˚
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 ˚.
Example Question #8 : How To Find The Angle Of A Sector
The area of the shaded sector in circle O is . What is the angle measure , rounded to the nearest hundredth?
˚
˚
˚
˚
˚
˚
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 area of the circle.
The area 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 ˚.
Example Question #9 : How To Find The Angle Of A Sector
The area of the shaded sector in circle O is . What is the angle measure ?
˚
˚
˚
˚
˚
˚
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 area of the circle.
The area 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 ˚.