A triangle has side lengths of 12 cm, 15 cm, and 20 cm. Classify it as acute, obtuse, or right.
To determine the type of triangle, we need to examine the relationship between the side lengths.
In a right triangle, one of the angles is a right angle (90 degrees) and follows the Pythagorean theorem (a^2 + b^2 = c^2) where 'c' is the hypotenuse (the longest side).
Let's check if this triangle follows the Pythagorean theorem:
12^2 + 15^2 = 144 + 225 = 369
20^2 = 400
As 369 is not equal to 400, the triangle is not a right triangle.
In an acute triangle, all three angles are less than 90 degrees. In an obtuse triangle, one of the angles is greater than 90 degrees.
To determine if the triangle is acute or obtuse, we can use the Law of Cosines:
a^2 = b^2 + c^2 - 2bc*cos(A)
b^2 = a^2 + c^2 - 2ac*cos(B)
c^2 = a^2 + b^2 - 2ab*cos(C)
Calculating the cosines of all angles, we find that the angle opposite the side length of 20 cm is approximately 133 degrees, which means the triangle is obtuse.