To determine the classification of a triangle with side lengths 10 inches, 12 inches, and 15 inches, we need to understand how to classify triangles based on the squares of their side lengths.
Let's consider the sides [tex]\(a = 10\)[/tex] inches, [tex]\(b = 12\)[/tex] inches, and [tex]\(c = 15\)[/tex] inches.
First, we compare these side lengths to see which is the largest. The largest side in this case is [tex]\(c = 15\)[/tex] inches.
Next, we need to calculate the squares of these side lengths:
- [tex]\(a^2 = 10^2 = 100\)[/tex]
- [tex]\(b^2 = 12^2 = 144\)[/tex]
- [tex]\(c^2 = 15^2 = 225\)[/tex]
With these calculations, we compare the sum of the squares of the two shorter sides ([tex]\(a\)[/tex] and [tex]\(b\)[/tex]) with the square of the longest side ([tex]\(c\)[/tex]):
1. Check if [tex]\(a^2 + b^2\)[/tex] compared to [tex]\(c^2\)[/tex]:
[tex]\[a^2 + b^2 = 100 + 144 = 244\][/tex]
[tex]\[c^2 = 225\][/tex]
Since:
[tex]\[a^2 + b^2 > c^2\][/tex]
[tex]\[244 > 225\][/tex]
This type of comparison indicates that the triangle is acute because the sum of the squares of the two shorter sides is greater than the square of the longest side.
2. Further confirmation:
Let's also verify by comparing [tex]\(b\)[/tex] and [tex]\(c\)[/tex] with [tex]\(a\)[/tex]:
[tex]\[b^2 + c^2 = 144 + 225 = 369\][/tex]
[tex]\[a^2 = 100\][/tex]
Since:
[tex]\[b^2 + c^2 > a^2\][/tex]
[tex]\[369 > 100\][/tex]
This again suggests the triangle is acute because the sum of the squares of the two sides, other than the smallest side, is greater than the square of the smallest side.
With both comparisons providing consistent results, we can confidently categorize the triangle correctly.
Thus, the correct classifications are:
- acute, because [tex]\(10^2 + 12^2 > 15^2\)[/tex]
- acute, because [tex]\(12^2 + 15^2 > 10^2\)[/tex]
