To classify a triangle with side lengths 6 cm, 10 cm, and 12 cm, we need to consider the relationship between the squares of the side lengths based on the Pythagorean theorem. Here's the step-by-step solution:
1. Identify the side lengths:
- [tex]\( a = 6 \)[/tex] cm
- [tex]\( b = 10 \)[/tex] cm
- [tex]\( c = 12 \)[/tex] cm
2. Calculate the squares of the side lengths:
- [tex]\( a^2 = 6^2 = 36 \)[/tex]
- [tex]\( b^2 = 10^2 = 100 \)[/tex]
- [tex]\( c^2 = 12^2 = 144 \)[/tex]
3. Compare the sum of the squares of the two shorter sides to the square of the longest side:
- Sum of the squares of the shorter sides: [tex]\( a^2 + b^2 = 36 + 100 = 136 \)[/tex]
- Square of the longest side: [tex]\( c^2 = 144 \)[/tex]
4. Classify the triangle based on the comparison:
- If [tex]\( a^2 + b^2 = c^2 \)[/tex], the triangle is a right triangle.
- If [tex]\( a^2 + b^2 > c^2 \)[/tex], the triangle is an acute triangle.
- If [tex]\( a^2 + b^2 < c^2 \)[/tex], the triangle is an obtuse triangle.
In our case:
- [tex]\( a^2 + b^2 = 136 \)[/tex]
- [tex]\( c^2 = 144 \)[/tex]
- Since [tex]\( 136 < 144 \)[/tex], the triangle is an obtuse triangle.
Therefore, the best 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].
