A quadrilateral P has (1, 1), (3, 1), (3, 5) and (1, 5) as 4

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed Jan 08, 2020 11:37 pm

[GMAT math practice question]

A quadrilateral P has (1, 1), (3, 1), (3, 5) and (1, 5) as 4 vertices and another quadrilateral Q has (-1, -1), (-5, -1), (-5, -5) and (-1, -5) as 4 vertices. A line divides these two quadrilaterals evenly at the same time. What is this line?

A. y = - 1/6x + 5/6
B. y = - 5/6x + 1/6
C. y = 6/5x + 3/5
D. y = 1/3x + 5/6
E. y = - 1/6x + 7/5

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Source: Beat The GMAT — Problem Solving | Wed Jan 08, 2020 11:37 pm

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by GMATGuruNY » Thu Jan 09, 2020 3:52 am

Max@Math Revolution wrote:[GMAT math practice question]

A quadrilateral P has (1, 1), (3, 1), (3, 5) and (1, 5) as 4 vertices and another quadrilateral Q has (-1, -1), (-5, -1), (-5, -5) and (-1, -5) as 4 vertices. A line divides these two quadrilaterals evenly at the same time. What is this line?

A. y = - 1/6x + 5/6
B. y = - 5/6x + 1/6
C. y = 6/5x + 3/5
D. y = 1/3x + 5/6
E. y = - 1/6x + 7/5

Each quadrilateral is a rectangle.
To divide a rectangle in half, a line must pass through the CENTER of the rectangle.

Center of P = (midpoint of the x-values, midpoint of the y-values) = ( (1+3)/2, (1+5)/2 ) = (2, 3)
Center of Q = (midpoint of the x-values, midpoint of the y-values) = ( (-1+(-5))/2, (-1+(-5))/2 ) = (-3, -3)

The correct answer must pass through the two centers (2, 3) and (-3, -3).
Slope of the line that passes through (2, 3) and (-3, -3) = ∆y/∆x = (-3-3)/(-3-2) = 6/5

The correct answer is C.

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

Re: A quadrilateral P has (1, 1), (3, 1), (3, 5) and (1, 5) as 4

by Max@Math Revolution » Sun Jan 12, 2020 5:44 pm

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The quadrilaterals are rectangles, and bisecting lines of rectangles pass through the center of the rectangles.
Thus we have to find the line passing through the centers of those two rectangles.
The centers of the rectangles are (2, 3) and (-3, -3).
The slope of the line passing through (2, 3) and (-3, -3) is (3- (-3)) / (2 - (-3)) = 6/5.
The line passing through them is y – 3 = (6/5)(x - 2) or y = (6/5)x + (3/5).
Therefore, C is the answer.
Answer: C

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