Cutting a triangle
See also Pyramid into fourths
Task.
Cut a 5 by 8 index card into two right triangles. Cut each triangle into 4 strips of equal area parallel to one of the edges of the card.
Plan.
The ratio of the areas of two similar triangles is proportional to the square of the ratios of their sides. (We will prove this below!)
Let W be a (whole) triangle and P be a part of W similar to W.
Thus x = (√(P/W))*a.
Why? And why do we need this to solve the problem? See below.
So the three cuts that divide a triangle with side a into four parts of equal areas should be the following distances from the corner.
0.5 * a, √(0.5) * a, and √(0.75) * a.
Thus, for a = 5, the cuts are 2.5 in., 3 1/2 in., and 4 5/16 in. from the corner.
And for a = 8, 4 in., 5 11/16 in., and 6 15/16 in. from the corner.
It is important that students carry out the plan with precision, cutting the two triangles into strips.
Remark. This problem can also be solved using calculus. Do you know how? Do you want to see it?
Cutting a triangle using calculus
We show only the case for the triangle that is 8 inches long and 5 inches high. We place the triangle in the coordinate plane as shown:
We integrate the function y = 5/8x as follows:
