Determine whether the measures 2,  8‾√,8, and  12‾‾‾√12 can be the measures of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. Justify your answer.

To determine whether these measures can form a triangle, we need to check the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given sides are 2, √8, and √12.

Let's first simplify √8 and √12.

√8 = √(4*2) = √4 * √2 = 2√2 ≈ 2.83 (rounded to two decimal places)

√12 = √(4*3) = √4 * √3 = 2√3 ≈ 3.46 (rounded to two decimal places)

Now, let's check the triangle inequality theorem:

2 + 2.83 > 3.46 (4.83 > 3.46) - True 2 + 3.46 > 2.83 (5.46 > 2.83) - True 2.83 + 3.46 > 2 (6.29 > 2) - True

Since all these inequalities are true, the measures 2, √8, and √12 can form a triangle.

Next, to classify the triangle as acute, obtuse, or right, we use the Pythagorean theorem. For a right triangle, the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the lengths of the other two sides. If it's greater, the triangle is obtuse. If it's less, the triangle is acute.

Let's check:

(2√3)^2 = (2)^2 + (2√2)^2 12 = 4 + 8 12 = 12

Since the two sides are equal, the triangle is a right triangle.