To determine the classification of the triangle with side lengths 6 cm, 10 cm, and 12 cm, follow these steps:
1. Identify the sides:
The sides of the triangle are given as [tex]\( a = 6 \)[/tex] cm, [tex]\( b = 10 \)[/tex] cm, and [tex]\( c = 12 \)[/tex] cm.
2. Determine if the given lengths satisfy the triangle inequality theorem:
To form a triangle, the sum of any two sides must be greater than the third side:
- [tex]\( 6 + 10 > 12 \)[/tex]
- [tex]\( 6 + 12 > 10 \)[/tex]
- [tex]\( 10 + 12 > 6 \)[/tex]
All these inequalities hold true, so the sides can form a triangle.
3. Check for obtuse triangle condition:
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. The longest side is 12 cm.
- Calculate the square of each side:
[tex]\[
6^2 = 36, \quad 10^2 = 100, \quad 12^2 = 144
\][/tex]
- Check if the square of the longest side is greater than the sum of the squares of the other two sides:
[tex]\[
12^2 > 6^2 + 10^2
\][/tex]
This means:
[tex]\[
144 > 36 + 100
\][/tex]
[tex]\[
144 > 136
\][/tex]
Since [tex]\( 144 \)[/tex] is indeed greater than [tex]\( 136 \)[/tex], this confirms that the triangle is obtuse.
Therefore, the correct classification for the triangle with side lengths 6 cm, 10 cm, and 12 cm is:
obtuse, because [tex]\( 6^2 + 10^2 < 12^2 \)[/tex].
