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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]\(12 \, \text{cm}\)[/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]\(6 + 10 \ \textgreater \ 12\)[/tex]

Sagot :

GinnyAnswer
Let's carefully examine the given triangle with side lengths of 6 cm, 10 cm, and 12 cm to determine its classification.

1. Identify and Check Sides:
- We have the side lengths: [tex]\( a = 6 \)[/tex] cm, [tex]\( b = 10 \)[/tex] cm, and [tex]\( c = 12 \)[/tex] cm.
- We consider [tex]\( c = 12 \)[/tex] cm as the longest side, which we will compare against the combination of squares of the other two sides.

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. Analyze Using the Pythagorean Theorem:
- Since [tex]\( a^2 + b^2 = 36 + 100 = 136 \)[/tex]
- Compare this sum with [tex]\( c^2 \)[/tex]:
- Notice that [tex]\( a^2 + b^2 \)[/tex] (136) is less than [tex]\( c^2 \)[/tex] (144).

4. Determine Triangle Type Based on Squares Comparison:
- According to triangle classification rules:
- If [tex]\( a^2 + b^2 = c^2 \)[/tex], the triangle is right-angled.
- If [tex]\( a^2 + b^2 > c^2 \)[/tex], the triangle is acute.
- If [tex]\( a^2 + b^2 < c^2 \)[/tex], the triangle is obtuse.
- Here, [tex]\( a^2 + b^2 < c^2 \)[/tex], so the triangle is obtuse.

5. Conclusion:
- The best classification for this triangle is "obtuse" because the sum of the squares of the two shorter sides (36 and 100) is less than the square of the longest side (144).

Thus, the correct answer is:

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