March 2023

The trick to solving this problem is to realize that the red triangle and the triangle directly above it are similar triangles. The angles that are touching each other are congruent because they are vertical opposite angles. The two other angles of the triangles are congruent due to the alternate angles theorem (this is because the top and bottom lines of the square are parallel).

We know that the red triangle has a height that is twice as large as the height of its similar triangle because its base is twice as large (the red triangle’s base spans a whole side of the square while the similar triangle only spans half of the side of a square). The two heights of the triangle add up to equal a side of the square, as shown above, which can be connoted using the variable 3h.

Therefore the area for the square is (3h)^2 = 9h^2
The area for the red triangle is (1/2)(3h)(2h) = 3h^2

The fraction of the area of the square that the red triangle takes up is (3h^2)/(9h^2) = 1/3