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Which classification best represents a triangle with side lengths 6 cm, 10 cm, and 12 cm?

A. Acute, because [tex]\(6^2 + 10^2 \ \textless \ 12^2\)[/tex]
B. Acute, because [tex]\(6 + 10 \ \textgreater \ 12\)[/tex]
C. Obtuse, because [tex]\(6^2 + 10^2 \ \textless \ 12^2\)[/tex]
D. Obtuse, because [tex]\(6 + 10 \ \textgreater \ 12\)[/tex]



Answer :

GinnyAnswer
To determine the classification of a triangle with side lengths [tex]\(6 \, \text{cm}\)[/tex], [tex]\(10 \, \text{cm}\)[/tex], and [tex]\(12 \, \text{cm}\)[/tex], we need to use the properties of triangle sides and the Pythagorean theorem.

### Step-by-Step Solution:

1. Calculate the squares of the side lengths:
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 10^2 = 100 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]

2. Use the Pythagorean theorem to check the type of triangle:
- For an obtuse triangle, the square of the longest side should be greater than the sum of the squares of the other two sides.
[tex]\[ 6^2 + 10^2 < 12^2 \][/tex]
[tex]\[ 36 + 100 < 144 \][/tex]
[tex]\[ 136 < 144 \][/tex]
This inequality is true, indicating that the triangle is obtuse.

3. Therefore, the classification that best represents the triangle is:
[tex]\[ \text{obtuse, because } 6^2 + 10^2 < 12^2 \][/tex]

### Conclusion:
The triangle with side lengths [tex]\(6 \, \text{cm}\)[/tex], [tex]\(10 \, \text{cm}\)[/tex], and [tex]\(12 \, \text{cm}\)[/tex] is best classified as an obtuse triangle. The correct statement is:

[tex]\[ \text{Obtuse, because } 6^2 + 10^2 < 12^2 \][/tex]

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