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Which classification best represents a triangle with side lengths [tex]$6 \, \text{cm}$[/tex], [tex]$10 \, \text{cm}$[/tex], and [tex][tex]$12 \, \text{cm}$[/tex][/tex]?

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][tex]$6 + 10 \ \textgreater \ 12$[/tex][/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], follow these steps:

1. Calculate the squares of the side lengths:
- For the first side: [tex]\(6^2 = 36\)[/tex]
- For the second side: [tex]\(10^2 = 100\)[/tex]
- For the third side: [tex]\(12^2 = 144\)[/tex]

2. Check the sums of the squares:
- Calculate [tex]\(6^2 + 10^2 = 36 + 100 = 136\)[/tex]
- Compare this sum to [tex]\(12^2 = 144\)[/tex]

3. Determine the triangle type based on the comparison:
- Notice that [tex]\(36 + 100 = 136\)[/tex] and [tex]\(136 < 144\)[/tex].

4. Conclusion:
- The inequality [tex]\(6^2 + 10^2 < 12^2\)[/tex] indicates that the triangle is obtuse (since in an obtuse triangle, the square of one side is greater than the sum of the squares of the other two sides).

Therefore, the classification that best represents the triangle is:

Obtuse, because [tex]\(6^2 + 10^2 < 12^2\)[/tex].

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