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The figure shows line segment PQ and a circle with radius 1
[#permalink]
08 Mar 2018, 14:24

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Attachment:

#GREpracticequestion The figure shows line segment PQ .jpg [ 12.23 KiB | Viewed 12250 times ]

The figure shows line segment PQ and a circle with radius 1 and center (5, 2) in the xy-plane. Which of the following values could be the distance between a point on line segment PQ and a point on the circle?

Indicate \(all\) such values.

A 2.5

B 3.0

C 3.5

D 4.0

E 4.5

F 5.0

G 5.5

H 6.0

Kudos to the right solution and explanation

Re: The figure shows line segment PQ and a circle with radius 1
[#permalink]
08 Mar 2018, 23:22

1

1

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B through F. Thorough explanation when I wake up in the morning haha

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Re: The figure shows line segment PQ and a circle with radius 1
[#permalink]
09 Mar 2018, 16:01

4

Expert Reply

Assuming the center of the circle is meant to be (5, 2), we should find both the minimum distance and the maximum distance between the line and the circle. The correct answers should be those two distances and everything in between.

Here's a good rule to know: the minimum distance of anything to a circle should be a straight line from the point to the center of the circle. This line will always, by the way, intercept the circle at a right angle. Anyway, in this problem we can see that drawing a line from the top/bottom of the line to the center of the circle will be longer than if we draw a line from the center of line PQ to the center of the circle. (This is because if we do both, you can see that the line from the top/bottom would be the hypotenuse of a right triangle, while the center of the line would be forming a leg of the same triangle, and thus is shorter.)

Since the circle has a center at (5, 2) and a radius of 1, we know that the left-hand side of the circle is at (4, 2) and a line from point (1, 2) to (4, 2) will have a length of 3. Thus, A is out, B is in, and anything larger than B is potentially in.

What's the maximum length? This is a tougher question. But since we've seen that starting at the top/bottom of line PQ makes the distance larger, let's start there. The easiest way to do this is to imagine that we are measuring the distance from the bottom of line PQ to the closest part of the circle. Then, the farthest part of the circle must be the exact opposite side of the circle. Since the closest side and the farthest side must be exactly 2 apart, (since the diameter is 2), we can just add 2 to this distance.

However, it's not super easy to find the closest part of the circle to point (1, 1), since we don't know exactly where it is. On the other hand, we do know exactly where the center of the circle is: (5, 2). We can find the distance from (1, 1) to (5, 2) by using the Pythagorean formula. The vertical distance of the right triangle formed by these two points is 1, and the horizontal distance is 4, so the Pythagorean formula tells us that:

1^2 + 4^2 = distance^2

so the distance is √17. This takes us to the center of the circle, but the far side must be another 1, since that's just adding the radius. So at this point we have the maximum distance as √17 + 1. What's √17 though? We can estimate it as a tiny bit more than 4, since √16 equals 4. So let's estimate the maximum distance as 5.1. After all, 5 squared is 25, and 17 is far closer to 16 than it is to 25. The next answer choice above 5.1 is 5.5, which is far too high. So F is our maximum at 5.

I would answer everything between B and F, inclusive. Why? If you can imagine the shortest possible line between the middle of line PQ and the left side of the circle, and then imagine smoothly sliding it all the way around the circle to the other side, we'll get an infinite set of distances until we reach the other side, which is a distance of 5. Then if we slide the side of the line that is on line PQ down to the bottom we'll increase the length by just a bit. But since we can slide it continuously we have a range of values.

_________________

Here's a good rule to know: the minimum distance of anything to a circle should be a straight line from the point to the center of the circle. This line will always, by the way, intercept the circle at a right angle. Anyway, in this problem we can see that drawing a line from the top/bottom of the line to the center of the circle will be longer than if we draw a line from the center of line PQ to the center of the circle. (This is because if we do both, you can see that the line from the top/bottom would be the hypotenuse of a right triangle, while the center of the line would be forming a leg of the same triangle, and thus is shorter.)

Since the circle has a center at (5, 2) and a radius of 1, we know that the left-hand side of the circle is at (4, 2) and a line from point (1, 2) to (4, 2) will have a length of 3. Thus, A is out, B is in, and anything larger than B is potentially in.

What's the maximum length? This is a tougher question. But since we've seen that starting at the top/bottom of line PQ makes the distance larger, let's start there. The easiest way to do this is to imagine that we are measuring the distance from the bottom of line PQ to the closest part of the circle. Then, the farthest part of the circle must be the exact opposite side of the circle. Since the closest side and the farthest side must be exactly 2 apart, (since the diameter is 2), we can just add 2 to this distance.

However, it's not super easy to find the closest part of the circle to point (1, 1), since we don't know exactly where it is. On the other hand, we do know exactly where the center of the circle is: (5, 2). We can find the distance from (1, 1) to (5, 2) by using the Pythagorean formula. The vertical distance of the right triangle formed by these two points is 1, and the horizontal distance is 4, so the Pythagorean formula tells us that:

1^2 + 4^2 = distance^2

so the distance is √17. This takes us to the center of the circle, but the far side must be another 1, since that's just adding the radius. So at this point we have the maximum distance as √17 + 1. What's √17 though? We can estimate it as a tiny bit more than 4, since √16 equals 4. So let's estimate the maximum distance as 5.1. After all, 5 squared is 25, and 17 is far closer to 16 than it is to 25. The next answer choice above 5.1 is 5.5, which is far too high. So F is our maximum at 5.

I would answer everything between B and F, inclusive. Why? If you can imagine the shortest possible line between the middle of line PQ and the left side of the circle, and then imagine smoothly sliding it all the way around the circle to the other side, we'll get an infinite set of distances until we reach the other side, which is a distance of 5. Then if we slide the side of the line that is on line PQ down to the bottom we'll increase the length by just a bit. But since we can slide it continuously we have a range of values.

_________________

Re: The figure shows line segment PQ and a circle with radius 1
[#permalink]
10 Mar 2018, 12:33

1

Expert Reply

Outstanding explanation Sir.

Regards

Regards

Re: The figure shows line segment PQ and a circle with radius 1
[#permalink]
08 Sep 2021, 10:58

Why can't we calculate distance between the top most point of circle (5,3) and the lowest point of line (1,1) ?

That would be 4 √ 5 = 8.94 . Which is the greatest distance. Implies that option G and H can be covered too .

The red line here - imgur . com /a/vCgxdZP

That would be 4 √ 5 = 8.94 . Which is the greatest distance. Implies that option G and H can be covered too .

The red line here - imgur . com /a/vCgxdZP

Re: The figure shows line segment PQ and a circle with radius 1
[#permalink]
08 Sep 2021, 11:31

Expert Reply

randomgreuser wrote:

Why can't we calculate distance between the top most point of circle (5,3) and the lowest point of line (1,1) ?

That would be 4 √ 5 = 8.94 . Which is the greatest distance. Implies that option G and H can be covered too .

The red line here - imgur . com /a/vCgxdZP

That would be 4 √ 5 = 8.94 . Which is the greatest distance. Implies that option G and H can be covered too .

The red line here - imgur . com /a/vCgxdZP

Please sir, fix the image

Re: The figure shows line segment PQ and a circle with radius 1
[#permalink]
14 Nov 2021, 12:07

hmmm

if the line segment is (1,3) and (1,1) then we look at the x values (in this case,1) to see how far right we have to go on the x axis to touch the circle. if the circle's center is 5,2 with radius 1 then we know the left part of the circle (closest part of the circle to the line) is (4,2) (aka 5(center)-radius 1=4 in the (4,2)). this means we have to travel from 1 on the x axis to 4 on the x axis moving 3. this eliminates anything under 3 so throw out A. similarly, if the center is (5,2) of the circle with radius 1 then the right part of the circle is 5+1 so (6,2) (aka 5(center)+radius 1=6 in the (6,2)) on the x axis. this means we can travel as far as 6 on the x axis after starting at 1 (6-1=5). so our boundaries are 3-5. keep b-f.

if the line segment is (1,3) and (1,1) then we look at the x values (in this case,1) to see how far right we have to go on the x axis to touch the circle. if the circle's center is 5,2 with radius 1 then we know the left part of the circle (closest part of the circle to the line) is (4,2) (aka 5(center)-radius 1=4 in the (4,2)). this means we have to travel from 1 on the x axis to 4 on the x axis moving 3. this eliminates anything under 3 so throw out A. similarly, if the center is (5,2) of the circle with radius 1 then the right part of the circle is 5+1 so (6,2) (aka 5(center)+radius 1=6 in the (6,2)) on the x axis. this means we can travel as far as 6 on the x axis after starting at 1 (6-1=5). so our boundaries are 3-5. keep b-f.

gmatclubot

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