Question
Is a triangle with side lengths 15, 10, and 12 a right triangle?
Answers
Answered by
GPT-4o mini
To determine if a triangle with side lengths 15, 10, and 12 is a right triangle, we can use the Pythagorean theorem. This theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the triangle is a right triangle if:
\[
a^2 + b^2 = c^2
\]
First, let's identify the sides. Here, the longest side is 15, so we have:
- \(a = 10\)
- \(b = 12\)
- \(c = 15\)
Now, we calculate \(a^2 + b^2\) and \(c^2\):
\[
a^2 + b^2 = 10^2 + 12^2 = 100 + 144 = 244
\]
\[
c^2 = 15^2 = 225
\]
Now we compare \(a^2 + b^2\) with \(c^2\):
\[
a^2 + b^2 = 244 \quad \text{and} \quad c^2 = 225
\]
Since \(244 \neq 225\), we conclude that:
\[
a^2 + b^2 \neq c^2
\]
Therefore, the triangle with sides 15, 10, and 12 is **not** a right triangle.
\[
a^2 + b^2 = c^2
\]
First, let's identify the sides. Here, the longest side is 15, so we have:
- \(a = 10\)
- \(b = 12\)
- \(c = 15\)
Now, we calculate \(a^2 + b^2\) and \(c^2\):
\[
a^2 + b^2 = 10^2 + 12^2 = 100 + 144 = 244
\]
\[
c^2 = 15^2 = 225
\]
Now we compare \(a^2 + b^2\) with \(c^2\):
\[
a^2 + b^2 = 244 \quad \text{and} \quad c^2 = 225
\]
Since \(244 \neq 225\), we conclude that:
\[
a^2 + b^2 \neq c^2
\]
Therefore, the triangle with sides 15, 10, and 12 is **not** a right triangle.