Question 1113878: Find the dimension of the largest rectangle that can be inscribed in the right triangle with sides 3, 4, and 5 if a side of the rectangle is on the hypotenuse of the triangle.
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website!  units
 units
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The Workout (assumes the max rectangle will have 1/2 the area of the 3-4-5 triangle, which I do not prove here, but it has been proven elsewhere):
Area of the 3-4-5 triangle is  sq units.
Area of largest rectangle is 1/2 that of the triangle, or 3 sq units.
If the triangle is drawn with side=3 along the y-axis, side=4 along the x-axis, and the hypotenuse connecting the points (0,3) with (4,0), then the corner of the rectangle that meets the x-axis is at (2,0) and the corner that meets the y-axis is at (0,3/2).
Thus the length of the rectangle is  units.
and the width of the rectangle is  units.
The picture below has blue lines drawn to help visualize the 3 pairs of congruent triangles. Exactly 3 of the 6 form the rectangle.
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