Answer:
the angles of the triangle are approximately 38.24 degrees, 67.53 degrees, and 74.23 degrees.
Explanation:
Let's use the Law of Cosines to solve this problem. According to this law, in any triangle with sides of lengths a, b, and c, and angles opposite those sides denoted by A, B, and C, respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
We can use this equation to find the cosine of the largest angle, which will allow us to determine the angles of the triangle.
In this case, the largest angle is opposite the longest side, which has a length of (x - 8) cm. So, let's rewrite the Law of Cosines equation in terms of x:
(x - 8)^2 = x^2 + (x - 4)^2 - 2x(x - 4)*cos(C)
Expanding and simplifying this equation gives:
x^2 - 16x + 64 = 2x^2 - 16x + 16 + 2x(x - 4)*cos(C)
Canceling out the common terms and rearranging, we get:
cos(C) = (x^2 + (x - 4)^2 - (x - 8)^2)/(2x(x - 4))
Simplifying the numerator and denominator gives:
cos(C) = (5x - 24)/(2x(x - 4))
Now, we can use a calculator to evaluate the cosine for different values of x and find the values that satisfy the given condition. Specifically, we want the cosine to be negative, since the largest angle of a triangle is always obtuse (greater than 90 degrees).
For example, if we try x = 10, we get:
cos(C) = (5(10) - 24)/(2(10)(6)) = -0.25
This satisfies the condition, so we can use this value of x to find the angles of the triangle:
Angle opposite x cm side = cos^-1((x^2 + (x - 4)^2 - (x - 8)^2)/(2x(x - 4)))
= cos^-1((10^2 + (10 - 4)^2 - (10 - 8)^2)/(210(10 - 4)))
= cos^-1(7/8)
≈ 38.24 degrees
Angle opposite (x - 4) cm side = cos^-1((x^2 + (x - 8)^2 - (x - 4)^2)/(2x(x - 8)))
= cos^-1((10^2 + (10 - 8)^2 - (10 - 4)^2)/(210(10 - 8)))
= cos^-1(3/8)
≈ 67.53 degrees
Angle opposite (x - 8) cm side = 180 - angle opposite x cm side - angle opposite (x - 4) cm side
= 180 - 38.24 - 67.53
≈ 74.23 degrees
Therefore, the angles of the triangle are approximately 38.24 degrees, 67.53 degrees, an the angles of the triangle are approximately 38.24 degrees, 67.53 degrees, and 74.23 degrees.
I hope this helped