Source: Revised GRE PDF 1st Ed. Section 6: Math; #25 (p. 91)

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# The figure shows line segment PQ and a

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. 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0

### 4 Explanations

1

Casey Caruso

How do we know that the line you drew is the farthest possible point? Couldn't a point under the (6,2) be even farther from point Q?

Dec 21, 2019 • Comment

Hi Casey,
Happy to help :)
Yes, we could draw a point farther from Q. But in order to be farther from Q, this point must be closer to P. And remember that we aren't looking for distances from Q, we're looking for distances from anywhere on the segment PQ. So the distance from Q (or P) to (6, 2) is the largest possible distance from a point on the circle to a point on the segment PQ. Anything farther from Q than that will be closer to P, and thus closer to a point on the segment PQ.
Hope that helps!

1

Kathleen M

This question confused me because it asks for which coordinates could be a point ON the circle. I assumed this meant on the circumference of the circle. How will I know if the question is asking for points on the edge of the circle or points within the circle?

Jul 19, 2019 • Comment

You got it: the questions specifies the distance between a point on the line segment and a point on the circumference of the circle. The inside of the circle, within it, does not count. "On" the circle means on the circumference of the circle. "In" the circle would mean inside or within it. But we need points on the circumference alone. So the distance should be between any point on the line segment and any point on the circumference of the circle.

3

Angela Sung

Why is the far edge of the circle 5, instead of 6? Isn't the center (5,2) with radius of 1?

Jun 2, 2013 • Comment

Lucas Fink, Magoosh Tutor

Good question, Angela! The far point of the circle is (6,2), it's true. However, that's only 5 away from the line segment QP at minimum, because QP is at the value of 1 on the x-axis. Since 6-1=5, the distance from the middle of QP to the far end of the circle is 5. I hope that helps :-)

2 Chris Lele, Magoosh Tutor

Sep 26, 2012 • Comment

Cristina Velasquez

Hello, Could you please explain how you knew that one of the points was 3.0 of distance away ? Thank you!

Lucas Fink, Magoosh Tutor

Christina— the point on the left side of the circle is 1 away from the center of the circle because the circle has a radius of 1. So that point would be (4,2). that is the closest part of the circle to the line, because it's the point on the circle that is leftmost.

Meanwhile, the point on the line that is closest to the circle is in the center of the line. If you move up or down the line, you can see that it moves away from the circle. That point in the center of the line is (1,2).

We know there is a distance of 3 from those two points— (4,2) (1,2)—by simple subtraction. The y coordinates are the same, so we only care about the 4 and the 1. The distance from 4 to 1 is 3 because 4-1 = 3.