Answer: :)
Explanation:
To classify the triangle as acute, right, or obtuse, we need to determine the measure of the largest angle in the triangle. We can use the Law of Cosines to find the angles of the triangle:
c^2 = a^2 + b^2 - 2ab cos(C)
where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.
For this triangle, we have:
c = 12
a = 6
b = 9
Using these values in the Law of Cosines, we get:
12^2 = 6^2 + 9^2 - 2(6)(9)cos(C)
144 = 117 - 108cos(C)
27 = 108cos(C)
cos(C) = 0.25
To find angle C, we can take the inverse cosine (cos^-1) of both sides:
C = cos^-1(0.25)
C ≈ 75.5°
Now we can classify the triangle based on its largest angle:
- If C is less than 90 degrees, then the triangle is acute.
- If C is exactly 90 degrees, then the triangle is right.
- If C is greater than 90 degrees, then the triangle is obtuse.
In this case, since C ≈ 75.5°, which is less than 90 degrees, we know that this triangle is an acute triangle.