To determine the classification of a triangle with side lengths [tex]\( 6 \)[/tex] cm, [tex]\( 10 \)[/tex] cm, and [tex]\( 12 \)[/tex] cm, we can use the relationship between the sides of a triangle and the angles within it, specifically the Pythagorean theorem. Here's the step-by-step process:
1. Check if it's a triangle: First, ensure that the side lengths can actually form a triangle. This can be checked using the triangle inequality theorem which states the sum of the lengths of any two sides must be greater than the length of the third side:
- [tex]\( 6 + 10 > 12 \)[/tex]: True
- [tex]\( 6 + 12 > 10 \)[/tex]: True
- [tex]\( 10 + 12 > 6 \)[/tex]: True
Since all these conditions are met, the side lengths do form a triangle.
2. Classify the triangle based on its angles:
- For a triangle to be acute, the square of the length of the longest side must be less than the sum of the squares of the other two sides:
[tex]\[c^2 < a^2 + b^2\][/tex]
- For a triangle to be right, the square of the length of the longest side must be equal to the sum of the squares of the other two sides:
[tex]\[c^2 = a^2 + b^2\][/tex]
- For a triangle to be obtuse, the square of the length of the longest side must be greater than the sum of the squares of the other two sides:
[tex]\[c^2 > a^2 + b^2\][/tex]
Here, [tex]\( c = 12 \)[/tex], [tex]\( a = 6 \)[/tex], and [tex]\( b = 10 \)[/tex]. We need to compare [tex]\( 12^2 \)[/tex] with [tex]\( 6^2 + 10^2 \)[/tex]:
- Compute [tex]\( 6^2 \)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
- Compute [tex]\( 10^2 \)[/tex]:
[tex]\[ 10^2 = 100 \][/tex]
- Compute [tex]\( 12^2 \)[/tex]:
[tex]\[ 12^2 = 144 \][/tex]
- Compare [tex]\( 12^2 \)[/tex] with [tex]\( 6^2 + 10^2 \)[/tex]:
\[ 144 \) (which is [tex]\( 12^2 \)[/tex])
\[ 36 + 100 = 136 \) (which is [tex]\( 6^2 + 10^2 \)[/tex])
Since [tex]\( 144 > 136 \)[/tex], the triangle with sides 6 cm, 10 cm, and 12 cm is classified as obtuse.
Hence, the correct classification of the triangle is:
- Obtuse, because [tex]\( 6^2 + 10^2 < 12^2 \)[/tex].
