Let's classify the triangle with the side lengths [tex]\( 6 \, \text{cm} \)[/tex], [tex]\( 10 \, \text{cm} \)[/tex], and [tex]\( 12 \, \text{cm} \)[/tex].
### Step 1: Calculate the squares of each side:
- For side [tex]\( a = 6 \)[/tex]:
[tex]\[
a^2 = 6^2 = 36
\][/tex]
- For side [tex]\( b = 10 \)[/tex]:
[tex]\[
b^2 = 10^2 = 100
\][/tex]
- For side [tex]\( c = 12 \)[/tex]:
[tex]\[
c^2 = 12^2 = 144
\][/tex]
Now, we have:
[tex]\[
a^2 = 36, \quad b^2 = 100, \quad c^2 = 144
\][/tex]
### Step 2: Sum of squares of the two smaller sides:
[tex]\[
a^2 + b^2 = 36 + 100 = 136
\][/tex]
### Step 3: Compare the sum of the squares with the square of the largest side:
We see that:
[tex]\[
a^2 + b^2 < c^2 \quad \text{(since } 136 < 144 \text{)}
\][/tex]
### Step 4: Conclusion based on the comparison:
- If [tex]\( a^2 + b^2 > c^2 \)[/tex], the triangle is acute.
- If [tex]\( a^2 + b^2 = c^2 \)[/tex], the triangle is right-angled.
- If [tex]\( a^2 + b^2 < c^2 \)[/tex], the triangle is obtuse.
Since [tex]\( 136 < 144 \)[/tex], we conclude that the triangle is obtuse.
Thus, the classification that best represents the triangle with side lengths [tex]\( 6 \, \text{cm} \)[/tex], [tex]\( 10 \, \text{cm} \)[/tex], and [tex]\( 12 \, \text{cm} \)[/tex] is:
[tex]\[
\boxed{\text{obtuse, because } 6^2 + 10^2 < 12^2}
\][/tex]
