Hey Amir,
Great question. One of my favorite things about coordinate geometry is that you can always find a right triangle, so that's the paradigm I use to approach these, and it usually works out pretty efficiently. My first take on this one was that, if that bottom angle (P) looks like a right angle, and if I could prove that it is I'll be able to use that base-height combination of those two lines that create it.
The advantage there is that those two lines both form right triangles with the x and y axes, so those lengths will be easy to calculate. They're actually both 3-4-5 triangles, so the length of each is 5.
Now, you can use the right triangle of that third side, which has a height of 1 and a length of 7, making its Pythagorean Theorem application:
1^2 + 7^2 = side^2
50 = side^2
Side = sqrt 50
Remember, my goal here was to prove that angle P was a right angle, which would make this problem significantly easier. If the isosceles sides are each 5, and the opposite side is sqrt 50, that fits with the 45-45-90 triangle ratio (x, x, sqrt2 * x), then we've proven that angle P is a right angle.
Because that's a right angle, we can use 5 as the base, 5 as the height, and the formula:
A = 1/2 bh = 1/2*5*5 = 12.5
When you're using coordinate geometry, use the fact that the whole coordinate grid is full of right angles - and therefore right triangles - to your advantage!
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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