# What is the short leg, the long leg, and the hypotenuse of the triangle?

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We can use the Pythagorean theorem:

`x^2+(1/2x+11)^2=(2x+1)^2`

`x^2+1/4x^2+11x+121=4x^2+4x+1`

`4x^2+x^2+44x+484=16x^2+16x+4`

`11x^2-28x-480=0`

`(11x+60)(x-8)=0`

By the zero product property `x=-60/11` or x=8.

As these are lengths, x must be positive.

So we take x=8.

**The side lengths are 8,15,17**

A triangle is a geometric object consisting of three sides (or legs). The sum of the interior angles in a triangle is 180 degrees. As a restriction (due to geometric construction), any leg must have a length less than the sum of the lengths of the other two legs (triangle inequality).

Triangles can be classified either by angle type or relative side length. By angle type, triangles can be classified as right, oblique, acute, obtuse, or degenerate. By relative side lengths, they can be classified as equilateral, isosceles, or scalene.

- Equilateral triangles are triangles whose sides all have equal lengths.
- Scalene triangles have side all with different lengths.
- Isosceles triangles have two legs of the same length.

In all cases (except for the equilateral that has same lengths for all sides), the 'longest' side is the one opposite the largest angle, and the 'shortest' side the one opposite the smallest angle. The terms 'long', 'short', and 'hypotenuse' only makes sense when we deal of right triangles, or triangles that contain an right (90 degree) angle -- can be scalene or isosceles.

The hypotenuse is the longest leg and is always the side opposite the 90-degree angle. The short leg is the one opposite the smallest angle, and the last leg is the long leg (true for all, but especially for scalene triangles). Some right triangles are special - e.g. 30-60-90 triangle (also a scalene) and a 45-45-90 triangle (referred to as an isosceles right triangle). In the case of the 45-45-90 triangle which has two sides of the same length, the short and long leg cannot be identified, but the hypotenuse remains to be the one opposite the 90-degree angle and is still the longest leg.

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