Pythagorean problem

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Example (3): The altitude of a right triangle in a plane is 15 cm. Please see the following diagram. If h is hypotenuse and p is the perimeter, express h as a function of p. 

Answer:

Using Pythagorean theorem,

Equations [1] and [2] have the same left hand sides. So, their right hand  sides must be the same.

Since b > 0, and p > h + 15, both left and right sides of this equation are positive. The function: y = square of x for  x > 0 is a one-to-one correspondence. So we do squaring operation on each side.

Since altitude = 15 cm, and h > 15 cm, perimeter p > 30cm. Hence, 2p – 30 > 0. Divide each side by 2p – 30 .

This is a rational function with degree of numerator greater than the degree of denominator by one. Hence, when p → ∞, then h(p) approaches to a linear function.

When p becomes indefinitely large (p → ∞), 

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