To determine the classification of a triangle with side lengths [tex]\(6 \, \text{cm}\)[/tex], [tex]\(10 \, \text{cm}\)[/tex], and [tex]\(12 \, \text{cm}\)[/tex], we can use the properties of the different types of triangles. Specifically, we want to verify if the triangle is acute, obtuse, or right.
### Step-by-Step Analysis
1. Identify the longest side:
- The side lengths are [tex]\(a = 6 \, \text{cm}\)[/tex], [tex]\(b = 10 \, \text{cm}\)[/tex], and [tex]\(c = 12 \, \text{cm}\)[/tex].
- The longest side is [tex]\(c = 12 \, \text{cm}\)[/tex].
2. Compare the square of the longest side with the sum of the squares of the other two sides:
- Calculate [tex]\(a^2\)[/tex], [tex]\(b^2\)[/tex], and [tex]\(c^2\)[/tex]:
[tex]\[
a^2 = 6^2 = 36
\][/tex]
[tex]\[
b^2 = 10^2 = 100
\][/tex]
[tex]\[
c^2 = 12^2 = 144
\][/tex]
- Add [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
[tex]\[
a^2 + b^2 = 36 + 100 = 136
\][/tex]
- Compare [tex]\(c^2\)[/tex] with [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
c^2 = 144
\][/tex]
[tex]\[
a^2 + b^2 = 136
\][/tex]
[tex]\[
144 > 136
\][/tex]
3. Determine the type of triangle:
- Since [tex]\(c^2 > a^2 + b^2\)[/tex], the triangle is obtuse. This is because for a triangle to be obtuse, the square of the longest side must be greater than the sum of the squares of the other two sides.
Therefore, the correct classification is:
Obtuse, because [tex]\(6^2 + 10^2 < 12^2\)[/tex].
