A triangle has side lengths of 18 cm, 80 cm, and 81 cm. Classify it as acute, obtuse, or right.
(1 point)
• obtuse
• right
• acute
To classify the triangle as acute, obtuse, or right, we need to determine if it satisfies the conditions for any of the three types of triangles.
In a right triangle, one of the angles measures exactly 90 degrees.
Let's use the Pythagorean theorem to determine if this triangle is a right triangle. According to the theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
We have side lengths of 18 cm, 80 cm, and 81 cm. To check if this triangle is right-angled, we can calculate \(18^2 + 80^2\) and check if the result is equal to \(81^2\).
\(18^2 + 80^2 = 324 + 6400 = 6724\)
\(81^2 = 6561\)
Since \(18^2 + 80^2\) is not equal to \(81^2\), we can conclude that the triangle is not a right triangle.
Now, let's determine if the triangle is obtuse or acute. In an obtuse triangle, one of the angles measures more than 90 degrees. In an acute triangle, all three angles measure less than 90 degrees.
To determine the angles, we can use the Law of Cosines, which states that in a triangle with side lengths a, b, and c and corresponding angles A, B, and C, the following relationship holds:
\(c^2 = a^2 + b^2 - 2ab\cos(C)\)
For our triangle, we can let a = 18 cm, b = 80 cm, and c = 81 cm. We can solve for the angle C (the angle opposite to side c) using the law of cosines:
\(81^2 = 18^2 + 80^2 - 2(18)(80)\cos(C)\)
\(6561 = 324 + 6400 - 2880\cos(C)\)
\(6561 = 6724 - 2880\cos(C)\)
\(-163 = -2880\cos(C)\)
\(\cos(C) = \frac{-163}{-2880} = \frac{163}{2880} \approx 0.0566\)
Since the cosine of an angle is positive when the angle is less than 90 degrees, we can conclude that \(C\) is acute.
Therefore, the triangle with side lengths of 18 cm, 80 cm, and 81 cm is classified as acute.