Final answer:
Option B. To determine if measures can be triangle sides, use the Triangle Inequality Theorem. The Pythagorean theorem then helps classify the triangle as acute, right, or obtuse by comparing the sum of the squares of the shorter sides to the square of the longest side.
Step-by-step explanation:
To determine whether a set of measures can be the sides of a triangle, you can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. For example, if you are given side lengths of 7, 10, and 12, you can check: 7 + 10 > 12, 7 + 12 > 10, and 10 + 12 > 7. Since all these conditions are met, these measures can form the sides of a triangle.
To classify the triangle as acute, right, or obtuse, you can use the Pythagorean theorem, which is a² + b² = c² for right triangles, where c is the hypotenuse (the longest side of the triangle). 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 acute. If it is less, the triangle is obtuse. For example, for the side lengths 3, 4, and 5, we check: 3² + 4² = 5². Since 9 + 16 equals 25, the triangle is right.