Sure, let's classify the triangle with side lengths of 6 cm, 10 cm, and 12 cm.
Step-by-step solution:
1. Understand the triangle inequalities: To classify a triangle by its angles, we often compare the squares of the side lengths.
2. Applying the Pythagorean Theorem:
- The side lengths are [tex]\( a = 6 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = 12 \)[/tex].
- Calculate the squares of these sides:
[tex]\[ a^2 = 6^2 = 36 \][/tex]
[tex]\[ b^2 = 10^2 = 100 \][/tex]
[tex]\[ c^2 = 12^2 = 144 \][/tex]
3. Analyze the relationship with these squares:
- Check the relationship between [tex]\( a^2 + b^2 \)[/tex] and [tex]\( c^2 \)[/tex]:
[tex]\[ a^2 + b^2 = 36 + 100 = 136 \][/tex]
[tex]\[ c^2 = 144 \][/tex]
- Compare [tex]\( a^2 + b^2 \)[/tex] to [tex]\( c^2 \)[/tex]:
[tex]\[ 136 < 144 \][/tex]
4. Classify based on the comparison:
- If [tex]\( a^2 + b^2 = c^2 \)[/tex], it is a right triangle.
- If [tex]\( a^2 + b^2 > c^2 \)[/tex], it is an acute triangle.
- If [tex]\( a^2 + b^2 < c^2 \)[/tex], it is an obtuse triangle.
In this case:
[tex]\[ 136 < 144 \][/tex]
Hence, this triangle is obtuse.
5. Conclusion:
The classification that represents this triangle is:
[tex]\[
\textbf{Obtuse, because } 6^2 + 10^2 < 12^2
\][/tex]
