Math CentralQuandaries & Queries


Question from Kristina, a student:

A square with side lengths of 6 cm is divided into 3 right triangles and a larger isosceles triangle. If the three right triangles have equal area, find the exact area of the isosceles triangle.


Try this shape.

triangle in a square

Stephen La Rocque.

If you need more help, scroll down.








Determine the three variable relationships:

This diagram labels the lengths that we need to consider as well as the area Q of the large isosceles triangle. We have three variables, so we need three distinct equations to solve the problem. They are:

   (1) Side length: x + y = 6

   (2) Equal areas: ½x2 = 3y

   (3) Total area: ½x2 + 3y + 3y + Q = 62.

Scroll down if you still need more hints.

















Get rid of y:
Combine the first two equations using the substitution method:

½x2 = 3y       x + y = 6
x2 - 6y = 0       y = 6 - x
x2 - 6(6 - x) = 0
x2 + 6x - 36 = 0


Now use the Quadratic Formula to solve for x. Since x is a length, we can ignore the negative solution and choose just the positive solution.

Scroll down for more of the solution.











Quadratic formula:

Next, we can combine equations (1) and (3) to solve for Q, the area we have been asked to find:

½x2 + 3y + 3y + Q = 62       x+y=6
Q = 36 - ½x2 - 6y       y=6-x
Q = 36 - ½x2 - 6(6-x)
Q = 6x - ½x2

And finally, substitute the value for x that we calculated earlier into this expression.










Final calculation:


Solved. You can check by calculating y and the area of each triangle and seeing that they add to 36 square centimeters.



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