To determine whether triangle RST with side lengths of 8 centimeters, 10 centimeters, and 13 centimeters is a right triangle, we will use the Pythagorean Theorem.
The Pythagorean Theorem states that in a right-angled triangle with the legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex] and the hypotenuse [tex]\(c\)[/tex], the following relationship holds:
[tex]\[a^2 + b^2 = c^2\][/tex]
To apply the theorem, we first need to identify the sides. Typically, in the context of the Pythagorean Theorem, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] will be the two shorter sides (legs), and [tex]\(c\)[/tex] will be the longest side (hypotenuse). For triangle RST, we can set:
- [tex]\(a = 8\)[/tex] cm
- [tex]\(b = 10\)[/tex] cm
- [tex]\(c = 13\)[/tex] cm
Next, we calculate [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
[tex]\[a^2 = 8^2 = 64\][/tex]
[tex]\[b^2 = 10^2 = 100\][/tex]
Then, we sum these values to find [tex]\(a^2 + b^2\)[/tex]:
[tex]\[a^2 + b^2 = 64 + 100 = 164\][/tex]
We then calculate [tex]\(c^2\)[/tex]:
[tex]\[c^2 = 13^2 = 169\][/tex]
To determine if triangle RST is a right triangle, we compare [tex]\(a^2 + b^2\)[/tex] with [tex]\(c^2\)[/tex]:
- [tex]\(a^2 + b^2 = 164\)[/tex]
- [tex]\(c^2 = 169\)[/tex]
Since [tex]\(164 \neq 169\)[/tex], we can conclude that:
[tex]\[a^2 + b^2 \neq c^2\][/tex]
Therefore, based on the Pythagorean Theorem, triangle RST is not a right triangle.
