If a triangle has sides of lengths 5, 12, and 13, what type of triangle is it?

To determine the type of triangle with side lengths 5, 12, and 13, we can use the Pythagorean theorem, which helps us identify whether a triangle is right-angled. According to this theorem, if a triangle has sides of lengths (a), (b), and (c) (where (c) is the longest side), then the triangle is a right triangle if the following equation holds true:

[ a^2 + b^2 = c^2 ]

For the given triangle, we can assign the sides as follows: (a = 5), (b = 12), and (c = 13). Now we calculate (a^2 + b^2) and (c^2):

1. Calculate (a^2 + b^2):

[

5^2 + 12^2 = 25 + 144 = 169

]

2. Calculate (c^2):

[

13^2 = 169

]

Since (a^2 + b^2 = c^2) holds true ((169 = 169)), we conclude