| Rectangle In A Triangle |
A rectangle is placed inside an isoceles right triangle in such a way that the two vertices of the rectangle lie on the hypotenuse, and the other two vertices lie on the legs.
The area of the triangle is 2 square units, and the area of the rectangle is one quarter of that.
In the diagram thus created, find the area of the triangle which does not touch the hypotenuse of the larger, isoceles triangle.
Note: For clarity, this problem has been reworded to say 'hypotenuse of the larger, isoceles triangle', rather than 'base of the larger, isoceles triangle'. |
Problem Moderated by: Douglas |
| Problem Solution |
The diagram below shows the setup. Note that this shows the longer side of the rectangle against the hypotenuse; there is another possibility, which will come out in the algebra.
The area of the rectangle must be 1/2, so we have the following quadratic:
(2√2 - 2x)x = 1/2, or
4x2 - 4√2x + 1 = 0
This works out to x = (√2 + 1)/2 or (√2 - 1)/2
The area of the triange which does not touch the base is:
area of the whole triangle - area rectangle - area of two small triangles.
= 2 - 1/2 - x2
Since there are two possible values for x, we must plug in the following values for x2:
(3 + 2√2)/4 and (3 - 2√2)/4
These give us final results of
(3 + 2√2)/4 and (3 - 2√2)/4
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