A nice little right-angled triangle
Geometry
Level
3
Given a right-angled triangle which area is the same as the length of the hypotenuse, let its 2 shorter sides be x and y.
When x+y is at its minimum, calculate the area of the triangle.
The answer is 4.
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1 solution
Let the triangle be ABC with the right angle at A, and P the foot of the perpendicular from A to BC. Then AP=2. Let BP=x. Then by similar triangles ABP and ACP, CP = x 4 . By AM-GM, the length of the hypotenuse, and hence the area, is x + x 4 ≥ 4 . Equality can be achieved in an isoceles right triangle.