To classify the given triangle with side lengths 10, 12, and 15, we will start by checking the properties of the sides in relation to each other. Specifically, we need to determine if the triangle is acute, right, or obtuse based on the square of the sides.
Given the sides are [tex]\(a = 10\)[/tex], [tex]\(b = 12\)[/tex], and [tex]\(c = 15\)[/tex], where [tex]\(c\)[/tex] is the longest side, we'll use the following criteria:
1. Acute Triangle: All angles are less than 90 degrees, and it satisfies [tex]\(a^2 + b^2 > c^2\)[/tex].
2. Right Triangle: One angle is exactly 90 degrees, and it satisfies [tex]\(a^2 + b^2 = c^2\)[/tex].
3. Obtuse Triangle: One angle is greater than 90 degrees, and it satisfies [tex]\(a^2 + b^2 < c^2\)[/tex].
Let's calculate the squares of the side lengths:
- [tex]\(a^2 = 10^2 = 100\)[/tex]
- [tex]\(b^2 = 12^2 = 144\)[/tex]
- [tex]\(c^2 = 15^2 = 225\)[/tex]
Now we check if the triangle satisfies the acute triangle condition:
[tex]\[a^2 + b^2 = 100 + 144 = 244\][/tex]
Since [tex]\(244 > 225\)[/tex], we have:
[tex]\[a^2 + b^2 > c^2\][/tex]
Therefore, since [tex]\(10^2 + 12^2 > 15^2\)[/tex], the given triangle is an acute triangle. Thus, the classification best representing the triangle is acute.
