a triangle has side lengths of 9cm, 18cm, and 12cm. Classify it as acute, obtuse, or right.
You have a scaled-up 3-4-6 triangle.
Since 3-4-5 is a right triangle, this one is obtuse because 6 > 5
To classify the given triangle as acute, obtuse, or right, we need to determine the measurement of its angles.
To do this, we can use the Law of Cosines. The law states that in any triangle with sides a, b, and c, and the angle opposite side a is denoted as A, then the following equation holds:
a^2 = b^2 + c^2 - 2bc * cos(A)
Let's calculate the three angles of the triangle:
1. Angle A:
Using the Law of Cosines, substituting a = 9cm, b = 18cm, and c = 12cm:
9^2 = 18^2 + 12^2 - 2 * 18 * 12 * cos(A)
81 = 324 + 144 - 432 * cos(A)
81 = 468 - 432 * cos(A)
-387 = -432 * cos(A)
cos(A) = -387 / -432
cos(A) ≈ 0.896
Since cosine is positive in the first and fourth quadrants, angle A is acute.
2. Angle B:
Using the same process, substituting a = 18cm, b = 9cm, and c = 12cm:
18^2 = 9^2 + 12^2 - 2 * 9 * 12 * cos(B)
324 = 81 + 144 - 216 * cos(B)
324 = 225 - 216 * cos(B)
-99 = -216 * cos(B)
cos(B) = -99 / -216
cos(B) ≈ 0.458
Since cosine is positive in the first and fourth quadrants, angle B is acute.
3. Angle C:
Since there are three angles in a triangle and their sum must be 180 degrees:
Angle C = 180 - (A + B)
Angle C = 180 - (≈ 57.52 + ≈ 79.84)
Angle C ≈ 42.64
Now that we know the three angles of the triangle, we can classify it:
If all angles are less than 90 degrees, the triangle is acute.
If one angle is exactly 90 degrees, the triangle is right.
If one angle is greater than 90 degrees, the triangle is obtuse.
In this case, all three angles of the triangle are less than 90 degrees. Therefore, the given triangle with side lengths 9cm, 18cm, and 12cm is an acute triangle.