coordinate geometry

[chacha0212]

[Brent@GMATPrepNow]

Let's draw a rectangle around the triangle (as shown below) and then subtract from the rectangle's area (28), the areas of the 3 right triangles that surround the triangle in question.

We get the following:


So, the area of PQR = 28 - (3.5 + 6 + 6) = 12.5

Answer = A

Cheers,
Brent

[GMATGuruNY]

In the rectangular coordinate system below, the are of triangular region PQR is

12.5
14
10√2
16
25

The area of the rectangle drawn around triangle PQR = 7*4 = 28.
Since triangle PQR takes up less than half the rectangle, PQR < 14.

The correct answer is A.

[ceilidh.erickson]

Brent and Mitch's explanations are both great approaches, but I just wanted to extrapolate a more general rule from this... in geometry, especially in area problems, the GMAT will often give you shapes that are difficult (or even impossible) to calculate on their own. In these cases, it's most helpful to think in terms of NEGATIVE SPACE: calculate the area of a larger shape that's easier to find, then subtract away the pieces that you don't want, leaving the area of the shape that you're trying to find.

[Mathsbuddy]

In this example, we can see that to travel from Q to P, we go [4,-3]
and to go from P to R, we go [3,4]
Hence line QP is perpendicular to PR.
In other words, QPR is a right-angle triangle.
Using 3-4-5, we also see that QP = PR = 5
So the area is simply 5x5/2 = 12.5

[Matt@VeritasPrep]

One last cool note about this problem. Suppose we have the triangle as shown below.



Since each of the other triangles is a 3-4-5, each other triangle must have angles of a and b, whatever those are. We know that a + b = 90, and we also know that 180 - (a+b) = the third angle of the blue triangle, since a, b, and that third angle are on a line.

Hence the blue triangle is a right triangle, since 180-(a+b) = 90°. That means that it has a base of 5 and a height of 5, and its area = 5*5/2 = 12.5.