2006 AMC 10B Problems/Problem 20
In rectangle , we have , , , for some integer . What is the area of rectangle ?
Let the slope of be and the slope of be .
Since and form a right angle:
Using the distance formula:
Therefore the area of rectangle is
This solution is the same as Solution 1 up to the point where we find that .
We build right triangles so we can use the Pythagorean Theorem. The triangle with hypotenuse has legs and , while the triangle with hypotenuse has legs and . Aha! The two triangles are similar, with one triangle having side lengths times the other!
Let . Then from our reasoning above, we have . Finally, the area of the rectangle is .