Given a hypotenuse of length 10 in a scalene triangle, what is the length of the largest leg?

To determine the length of the largest leg in a scalene triangle with a hypotenuse of length 10, we need to consider the properties of right triangles since the challenge involves maximizing one of the legs while adhering to the triangle inequality and the Pythagorean theorem.

In a right triangle, where the hypotenuse is the longest side, the lengths of the legs ( a ) and ( b ) must satisfy the equation ( a^2 + b^2 = c^2 ), where ( c ) is the hypotenuse. In this case, since the hypotenuse is given as 10, we have:

[

a^2 + b^2 = 10^2 = 100

]

To find the maximum possible length for the larger leg, we can assign ( a ) to be the leg we want to maximize. Thus, we can express ( b ) in terms of ( a ):

[

b^2 = 100 - a^2

]

Since we are seeking the largest leg, it makes sense that both ( a ) and ( b ) need to be positive, and we can deduce that for a valid triangle, both legs cannot