A triangle has side lengths of 18 cm, 80 cm, and 81 cm. Classify it as acute, obtuse, or right.
(1 point)
Responses
obtuse
obtuse
right
right
acute
acute
To determine whether a triangle is acute, obtuse, or right, we need to apply the Pythagorean Theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's calculate the squares of the given side lengths:
18^2 = 324
80^2 = 6400
81^2 = 6561
Now, we need to check if any of the side lengths satisfy the conditions of the Pythagorean Theorem. By comparing the squares, we can see that:
324 + 6400 = 6724 (not equal to 6561)
324 + 6561 = 6885 (not equal to 6400)
6400 + 6561 = 12961 (not equal to 324)
Since none of the sums of the squares of any two sides equal the square of the remaining side, we can conclude that this triangle is not a right triangle.
Now, we need to determine if the triangle is acute or obtuse. In an acute triangle, the sum of the squares of the two smaller sides is greater than the square of the longest side. In an obtuse triangle, the square of the longest side is greater than the sum of the squares of the other two sides.
Let's compare the sums of the squares:
324 + 6400 = 6724 (not greater than 6561)
324 + 6561 = 6885 (greater than 6400)
6400 + 6561 = 12961 (greater than 324)
From this comparison, we can see that the sum of the squares of the two smaller sides, 80 cm and 81 cm, is greater than the square of the longest side, 18 cm. Therefore, the triangle is classified as an obtuse triangle.