ISEE Upper Level Quantitative : Plane Geometry
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Example Questions
Example Question #161 : Geometry
The circumferences of eight circles form an arithmetic sequence. The smallest circle has radius two inches; the second smallest circle has radius five inches. Give the radius of the largest circle.
3 feet 10 inches
1 foot, 11 inches
2 feet
4 feet 2 inches
2 feet, 1 inch
1 foot, 11 inches
The circumference of a circle can be determined by multiplying its radius by 
and
The circumferences form an arithmetic sequence with common difference
The circumference of a circle can therefore be found using the formula
where 
Since the radius of a circle is the circumference of the circle divided by 
Example Question #162 : Geometry
The track at James Buchanan High School is shown above; it is comprised of a square and a semicircle.
Diane wants to run two miles. If she begins at Point A and begins running counterclockwise, when she is finished, which of the five points will she be closest to?
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is
The distance around the track is about
Diane wants to run two miles, or
She will make about
circuits around the track.
Equivalently, she will run the track 5 complete times for a total of about
so she will have
She is running counterclockwise, so she will proceed from Point A to Point D, running another 800 feet, leaving
She will almost, but not quite, finish the 628 feet from Point D to Point B.
The correct response is Point B.
Example Question #5 : How To Find Circumference
The track at Monroe Elementary School is a perfect circle of radius 400 feet, and is shown in the above figure.
Evan and his younger brother Mike both start running from Point A. Evan runs counterclockwise, running once around the track and then on to Point E; Mike runs clockwise, meeting Evan at Point E and stopping.
Which of the following is the greater quantity?
(a) Twice Mike's average speed.
(b) Evan's average speed.
(Assume the five points are evenly spaced)
(b) is greater
(a) is greater
(a) and (b) are equal
It is impossible to tell which is greater from the information given
(b) is greater
It is not actually necessary to know the radius or length of the track if we know the points are equally spaced. Evan runs once around the track counterclockwise and then on to Point E, which is the next point after A; this means he runs around the track 
Therefore, Evan's speed is 
Example Question #22 : Radius
The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure.
Jerry begins his one-mile run at Point A, then runs counterclockwise around the track. At the end of his one-mile run, which is the greater quantity?
(a) The additional distance he would have to run if he were to continue to run counterclockwise to Point A.
(b) The additional distance he would have to run if he were to turn back and run clockwise to Point A.
(a) and (b) are equal
It is impossible to tell from the information given
(b) is greater
(a) is greater
(a) is greater
The circumference of a circle with radius 600 feet is
A one mile run would be
times the length of the track.
Therefore, Jerry's run takes him around the track once, and about 0.4 times the length of the track. Since he is running counterclockwise, but has not made it halfway around the track yet, the longer of the two paths is to proceed counterclockwise and run the remaining 0.6 of the track. This makes (a) greater.
Example Question #7 : How To Find Circumference
The track at Simon Bolivar High School is a perfect circle of radius 500 feet, and is shown in the above figure. Manuel starts at point C, runs around the track counterclockwise three times, and continues to run clockwise until he makes it to point D. Which of the following comes closest to the number of miles Manuel has run?
The circumference of a circle with radius 500 feet is
Manuel runs this distance three times, then he runs from Point C to D, which is about four-fifths of this distance. Therefore, Manuel's run will be about
Divide by 5,280 to convert to miles:
making 
Example Question #31 : Circles
The radii of six circles form an arithmetic sequence. The radius of the second-smallest circle is twice that of the smallest circle. Which of the following, if either, is the greater quantity?
(a) The circumference of the largest circle
(b) Twice the circumference of the third-smallest circle
(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
Call the radius of the smallest circle 
The radii of the six circles are, from least to greatest:
The largest circle has circumference
The third-smallest circle has circumference:
Twice this is
The circumference of the sixth circle is equal to twice that of the third-smallest circle, so the correct choice is that that (a) and (b) are equal.
Example Question #3 : How To Find Circumference
The track at Truman High School is shown above; it is comprised of a square and a semicircle.
Veronica begins at Point A, runs three times around the track counterclockwise, and continues until she reaches Point B. Which of the following comes closest to the distance Veronica runs?
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is about
The distance around the track is about
Veronica runs around the track three complete times, for a distance of about
She then runs from Point A to Point E, which is another 500 feet; Point E to Point D, which is yet another 500 feet, and, finally Point D to Point B, for a final 785 feet. The total distance Veronica runs is about
Divide by 5,280 to convert to miles:
The closest answer is 
Example Question #4 : How To Find Circumference
The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure. Quinnella wants to run around the track for one and a half miles. If Quinnella starts at point C and runs counterclockwise, which of the following is closest to the point at which she will stop running?
(Assume the five points are evenly spaced)
Between Points E and A
Between Points A and B
Between Points B and C
Between Points C and D
Between Points D and E
Between Points B and C
A circle of radius 600 feet will have a circumference of
Quinnella will run one and a half miles, or
which is about 
Quinnella will run around the track twice, returning to Point C; she will not quite make it to Point B a third time, since that is one-fifth of the track, or 0.2. The correct response is that she will be between Points B and C.
Example Question #161 : Geometry
In the above figure, 
Which is the greater quantity?
(a) The sum of the circumferences of the inner and outer circles
(b) The sum of the circumferences of the second-largest and third-largest circles
(a) and (b) are equal
(b) is the greater quantity
It is impossible to determine which is greater from the information given
(a) is the greater quantity
(a) and (b) are equal
For the sake of simplicity, we will assume that 
The four concentric circles have radii 1, 2, 3, and 4, respectively, and their circumferences can be found by multiplying these radii by 
The inner and outer circles have circumferences 
The two sums are therefore equal.
Example Question #32 : Circles
What is the angle measure of 
It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle. To do this, you merely need to multiply 
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