To determine the classification of a triangle with side lengths of 10 inches, 12 inches, and 15 inches, we need to use the properties of triangles and the Pythagorean theorem.
First, let's recall that the type of a triangle can be classified based on the relation between the squares of its sides:
- An acute triangle has all angles less than 90 degrees, which implies for the side lengths [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a^2 + b^2 > c^2, \quad b^2 + c^2 > a^2, \quad a^2 + c^2 > b^2
\][/tex]
- An obtuse triangle has one angle greater than 90 degrees, which implies for one pair of side lengths:
[tex]\[
a^2 + b^2 < c^2 \quad \text{or} \quad b^2 + c^2 < a^2 \quad \text{or} \quad a^2 + c^2 < b^2
\][/tex]
Given the side lengths:
- [tex]\( a = 10 \)[/tex]
- [tex]\( b = 12 \)[/tex]
- [tex]\( c = 15 \)[/tex]
Let's calculate the squares of these side lengths:
- [tex]\( 10^2 = 100 \)[/tex]
- [tex]\( 12^2 = 144 \)[/tex]
- [tex]\( 15^2 = 225 \)[/tex]
Now let's check the inequalities to classify the triangle:
1. Check if [tex]\( 10^2 + 12^2 > 15^2 \)[/tex]:
[tex]\[
100 + 144 > 225 \implies 244 > 225 \quad \text{(true)}
\][/tex]
Therefore, because [tex]\( 10^2 + 12^2 > 15^2 \)[/tex], this condition meets the criterion for an acute triangle.
2. Check if [tex]\( 12^2 + 15^2 > 10^2 \)[/tex]:
[tex]\[
144 + 225 > 100 \implies 369 > 100 \quad \text{(true)}
\][/tex]
Therefore, because [tex]\( 12^2 + 15^2 > 10^2 \)[/tex], this condition also meets the criterion for an acute triangle.
Since both conditions [tex]\( 10^2 + 12^2 > 15^2 \)[/tex] and [tex]\( 12^2 + 15^2 > 10^2 \)[/tex] hold true, we can confirm that the triangle is acute.
Hence, the classification of the triangle is:
- acute, because [tex]\( 10^2 + 12^2 > 15^2 \)[/tex]
- acute, because [tex]\( 12^2 + 15^2 > 10^2 \)[/tex]
Thus, the correct answers are:
1. "acute, because [tex]\( 10^2 + 12^2 > 15^2 \)[/tex]"
2. "acute, because [tex]\( 12^2 + 15^2 > 10^2 \)[/tex]"
