To classify a triangle based on the side lengths [tex]\(6 \, \text{cm}\)[/tex], [tex]\(10 \, \text{cm}\)[/tex], and [tex]\(12 \, \text{cm}\)[/tex], we can use the properties of triangles and the Pythagorean Theorem.
### Step 1: Verify Triangle Inequality Theorem
First, let's ensure these side lengths can form a triangle. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We need to check:
1. [tex]\(6 + 10 > 12\)[/tex]
2. [tex]\(6 + 12 > 10\)[/tex]
3. [tex]\(10 + 12 > 6\)[/tex]
Checking these:
1. [tex]\(6 + 10 = 16\)[/tex], which is greater than [tex]\(12\)[/tex].
2. [tex]\(6 + 12 = 18\)[/tex], which is greater than [tex]\(10\)[/tex].
3. [tex]\(10 + 12 = 22\)[/tex], which is greater than [tex]\(6\)[/tex].
Since all conditions are satisfied, the side lengths form a valid triangle.
### Step 2: Calculate the Squares of the Sides
Next, we compute the squares of the side lengths for use in the Pythagorean Theorem:
1. [tex]\(6^2 = 36\)[/tex]
2. [tex]\(10^2 = 100\)[/tex]
3. [tex]\(12^2 = 144\)[/tex]
### Step 3: Compare Sums of Squares to Classify the Triangle
To classify the triangle, we compare the sum of the squares of the two shorter sides to the square of the longest side:
1. Sum of squares of shorter sides: [tex]\(6^2 + 10^2 = 36 + 100 = 136\)[/tex]
2. Square of the longest side: [tex]\(12^2 = 144\)[/tex]
### Interpretation
According to the Pythagorean Theorem and its extensions:
1. If [tex]\(a^2 + b^2 = c^2\)[/tex], the triangle is right.
2. If [tex]\(a^2 + b^2 > c^2\)[/tex], the triangle is acute.
3. If [tex]\(a^2 + b^2 < c^2\)[/tex], the triangle is obtuse.
Here, we have [tex]\(6^2 + 10^2 = 136\)[/tex] and [tex]\(12^2 = 144\)[/tex]. Since [tex]\(136 < 144\)[/tex], we conclude that the triangle is obtuse.
### Conclusion
The classification that best represents a triangle with side lengths [tex]\(6 \, \text{cm}\)[/tex], [tex]\(10 \, \text{cm}\)[/tex], and [tex]\(12 \, \text{cm}\)[/tex] is:
- obtuse, because [tex]\(6^2 + 10^2 < 12^2\)[/tex].
So the correct answer is:
- obtuse, because [tex]\(6^2 + 10^2 < 12^2\)[/tex].
