Geometry

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prachi18oct: Master | Next Rank: 500 Posts; Posts: 269; Joined: Sun Apr 27, 2014 10:33 pmFollowed by: 5 members; Thanked: 8 times

by prachi18oct » Thu Jul 09, 2015 10:52 am

How many integer points lie between points A and B on the line segment AB, if A is (5, 7) and B is (10, -3)?
A)4
B)5
C)6
D)10
E)15

No clue how to proceed with this. Please help!

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Source: Beat The GMAT — Problem Solving | Thu Jul 09, 2015 10:52 am

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by [email protected] » Thu Jul 09, 2015 3:15 pm

Hi prachi18oct,

You might find it helpful to draw the two points and the line segment between them, but doing so is not necessary to get to the solution.

To figure out the number of integer points between point A and point B, we need to know the SLOPE of the line segment (from the answer choices, we know that there are from 4 to 15 integer points):

Slope = (Change in Y)/(Change in X) = Rise/Run

Point A = (5,7)
Point B = (10,-3)

Slope = (7 - (-3))/(5-10) = 10/-5 = -2

Since the slope is - 2, for every 1 we "move to the right", we move "down" 2.... we can now 'map out' the various integer co-ordinates between Point A and Point B:

Point A = (5,7)
(6, 5)
(7, 3)
(8, 1)
(9, -1)
Point B = (10, -3)

Since the question asks for the number of points BETWEEN A and B, the answer is 4.

Final Answer: A

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Rich

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prachi18oct: Master | Next Rank: 500 Posts; Posts: 269; Joined: Sun Apr 27, 2014 10:33 pmFollowed by: 5 members; Thanked: 8 times

by prachi18oct » Thu Jul 09, 2015 7:06 pm

[email protected] wrote:Hi prachi18oct,

Slope = (7 - (-3))/(5-10) = 10/-5 = -2

Since the slope is - 2, for every 1 we "move to the right", we move "down" 2.... we can now 'map out' the various integer co-ordinates between Point A and Point B:

GMAT assassins aren't born, they're made,
Rich

Can you explain the bolded part in detail.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Jul 09, 2015 7:32 pm

prachi18oct wrote:How many integer points lie between points A and B on the line segment AB, if A is (5, 7) and B is (10, -3)?
A)4
B)5
C)6
D)10
E)15

Slope = (yâ‚‚ - yâ‚�)/(xâ‚‚ - xâ‚�) = (-3-7)/(10-5) = -10/5 = -2.
Thus, the equation of the line is as follows:
y = -2x + b.

To determine the value of b, plug (x, y) = (5,7) into y = -2x + b and solve for b:
7 = -2*5 + b
b = 17.
Thus, the equation of the line is as follows:
y = -2x + 17.

Since A = (5, 7) and B = (10, -3), any point BETWEEN A AND B must have an x-coordinate BETWEEN 5 AND 10.
Between 5 and 10 there are four integer options for x:
6, 7, 8 and 9.
Plug x=6, x=7, x=8 and x=9 into y = -2x + 17.

If x=6, then y = -2x + 17 = -2*6 + 17 = 5.
If x=7, then y = -2x + 17 = -2*7 + 17 = 3.
If x=8, then y = -2x + 17 = -2*8 + 17 = 1.
If x=9, then y = -2x + 17 = -2*9 + 17 = -1.

Thus, four points between A and B have coordinates that are integer values:
(6, 5)
(7, 3)
(8, 1)
(9, -1).

The correct answer is A.

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