Final answer:
The side lengths 8, 12, and 9 can form a triangle. The triangle formed is acute.
Step-by-step explanation:
To determine if the side lengths will make a triangle, we need to check if the sum of the lengths of any two sides is greater than the length of the third side.
Let's apply this condition to the side lengths 8, 12, and 9:
8 + 12 = 20, which is greater than 9
12 + 9 = 21, which is greater than 8
9 + 8 = 17, which is less than 12
Since the sum of the lengths of any two sides is greater than the length of the third side for the side lengths 8, 12, and 9, they can form a triangle.
To classify the triangle, we can use the Pythagorean theorem. If the sum of the squares of the two shorter sides is equal to the square of the longest side, the triangle is right.
If the sum of the squares of the two shorter sides is greater than the square of the longest side, the triangle is obtuse. If the sum of the squares of the two shorter sides is less than the square of the longest side, the triangle is acute.
Let's calculate:
8^2 + 9^2 = 64 + 81
= 145
12^2 = 144
Since 64 + 81 is less than 144, the triangle with side lengths 8, 12, and 9 is acute.