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If you have run through the Co-ordinate Geometry module, then you must check out this question. It may be beyond GMAT level but it tests the GMAT relevant concepts in a most innovative way.

Question: Triangles and have areas 2007 and 7002, respectively, with , , , and ). What is the sum of all possible -coordinates of ?

(A) 282

(B) 300

(C) 600

(D) 900

(E) 1200

**Solution:**

The worse the numbers given, the more the probability that they will not get used at all.

Draw a diagram. lies on axis. To make a triangle with a given area (2007) i.e. a fixed altitude (whatever that may be, it is 18 in this case), will lie on any point on the two red lines. The two red lines are 18 units above and below the axis. Think about this before moving ahead.

Now draw . To make a triangle with a given area (7002) i.e. a fixed altitude (whatever that may be), will lie on any point on the two given dotted purple lines.

So what we need is the co-ordinates of the 4 possible values (, , and )

Now notice the co-ordinates of and are , and . So the diff between co-ordinates is 9 and between co-ordinates is also 9. This means is a line with slope 1.

Hence, it will cut the axis at 300 (when we move y co-ordinate 380 units down, we will move the co-ordinate 380 units to the left so the co-ordinate will become 680 – 380 = 300). This is the point .

Now note that slope of purple dotted lines is also 1 since they are parallel to .

By symmetry, the 4 possible co-ordinates of

When we add them, all and will get cancelled to give a total of 1200.

Answer **(E)**.

The entire calculation required was just but why to arrive at this expression was an altogether different game.