Question

One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse

Answer 1

Without loss of generality, we can think of the 4 cm leg of the triangle as being the short leg. It is also the hypotenuse of the smallest of the similar triangles created by the altitude (x). If h represents the hypotenuse of the triangle, then similarity tells us the ratio of the "long" leg to the hypotenuse is the same for the smallest and largest triangles.

... x/4 = √(h²-4²)/h

Then the altitude (x) as a function of the hypotenuse (h) is ...

... x = (4/h)×√(h²-16)

Answer 2

This is about division of a right angle triangle.

y =  $\frac{4}{h}$√(h² - 4²)

• To solve this question, i have drawn a triangle showing this leg of 4 cm and i have attached it.

• From the attachment, i have also drawn the altitude with its length denoted by y cm while the hypotenuse has its length denoted by h cm.

• Now, the length along the longer leg of the original triangle will be gotten from Pythagoras theorem. Thus;

Longer leg = √(h² - 4²)

• Now, in the smaller triangle we can see that the side with length as 4 is now the hypotenuse. From the concept of ratio of similar triangles, we can say that;

y/4 = (√(h² - 4²))/h

Thus, the altitude as a function of the Length of the hypotenuse gives;

y = $\frac{4}{h}\sqrt{h^{2} -4^{2} }$

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