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GMAT Coordinate Geometry
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Author:  questioner [ Tue Jun 08, 2010 3:57 pm ]
Post subject:  GMAT Coordinate Geometry

In a rectangular coordinate system, what is the area of a quadrilateral whose vertices have the coordinates (2,-2), (2, 6), (15, 2), (15,-4)?
A. 91
B. 95
C. 104
D. 117
E. 182

(A) First, we should make a rough sketch of the figure to determine its general shape. Its left side and right side are parallel, with the left side having a length of 8 and the right side having a length of 6. The distance between these two sides is 13.This figure is a trapezoid. A trapezoid is any quadrilateral that has one set of parallel sides, and the formula for the area of a trapezoid is:

Area = (1/2) × (Base 1 + Base 2) × (Height), where the bases are the parallel sides.

We can now determine the area of the quadrilateral:

Area = 1/2 × (8 + 6) × 13 = 1/2 × 14 × 13 = 7 × 13 = 91.

The correct answer is choice (A).

Alternate Method (Breaking the figure apart):
Without the formula for the area of a trapezoid, we can still solve the problem. We can draw two horizontal lines through the figure, one at y = 2 and one at y = -2 to divide the trapezoid
into an upper triangle, a rectangle, and a lower triangle.

The upper triangle has an area of (1/2) × 4 × 13 = 26.
The rectangle has an area of 4 × 13 = 52.
The lower triangle has an area of 1/2 × 2 × 13 = 13.

Adding these areas, we get the area for the quadrilateral:
52 + 26 + 13 = 91.

Again, we see that the correct answer is choice (A).
-------------

If the y coordinates are -2 and 6 then the length of the left side is 9 and the right side is 7 (you need to count the 0)!
Area = 1/2 × (9+7) × 13 = 104
(and not 91)

Author:  Gennadiy [ Wed Jun 09, 2010 4:55 pm ]
Post subject:  Re: GMAT Coordinate Geometry

Image

It may be easier if you break each segment in two.
Segment that connects points (2, -2) and (2, 6) we can break into
(2, 0) - (2, 6) segment, length 6
&
(2, -2) - (2, 0) segment, length 2.

The total length is 8.

The same works for the right side.

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