Question

a square is split into four triangles, and then three of the four triangles are shaded as shown. If the areas of the shaded triangles are 3,4, and 6, as shown, what is the area of the unshaded triangle?

Answer 1

11

Explanation

We first let the side of the square be defined as x.

Therefore, we have ax/2 = 4, and bx/2= 3. Where a and b are the legs of the triangles with areas 4 and 3 respectively.

For the Triangle with area 6, we then have ((x-a)(x-b))/2 = 6. Expanding gives us x^2-bx-ax+ab=12. Using the equation above and plugging in values where bx = 6, ax=8, a = 8/x, and b = 6/x. We get

x^2-6-8+48/x^2 = 12.

Simplified gives us,

x^4-26x^2+48.

Factored gives us,

((x^2-2)(x^2-24).

Since the area cannot be any negative values and bigger than 6 (since there is already a triangle with area 6), we then have x^2 = 24.

Therefore a of unshaded region = 24-6-4-3= 11