The perimeter of triangle ABC is 30 ft. AB = 3x+1, BC = 4x, and AC = 5x-7. List the angles of triangle ABC in order from smallest to largest. (Smallest on top, largest on bottom)(1 point)
To find the angles of triangle ABC, we need to find the values of x.
We know that the perimeter of a triangle is the sum of the lengths of its sides. So, we have:
AB + BC + AC = 30
Substituting the given expressions for AB, BC, and AC, we get:
3x + 1 + 4x + 5x - 7 = 30
Combining like terms, we have:
12x - 6 = 30
Adding 6 to both sides, we get:
12x = 36
Dividing both sides by 12, we obtain:
x = 3
Now we can substitute this value of x into the expressions for AB, BC, and AC to find their lengths:
AB = 3x + 1 = 3(3) + 1 = 10
BC = 4x = 4(3) = 12
AC = 5x - 7 = 5(3) - 7 = 8
Now we can use the lengths of the sides of the triangle to determine the angles.
Using the Law of Cosines, we have:
cos(A) = (BC^2 + AC^2 - AB^2) / (2 * BC * AC)
= (12^2 + 8^2 - 10^2) / (2 * 12 * 8)
= (144 + 64 - 100) / 192
= 108 / 192
= 9 / 16
cos(B) = (AB^2 + AC^2 - BC^2) / (2 * AB * AC)
= (10^2 + 8^2 - 12^2) / (2 * 10 * 8)
= (100 + 64 - 144) / 160
= 20 / 160
= 1 / 8
cos(C) = (AB^2 + BC^2 - AC^2) / (2 * AB * BC)
= (10^2 + 12^2 - 8^2) / (2 * 10 * 12)
= (100 + 144 - 64) / 240
= 180 / 240
= 3 / 4
Now we can use the inverse cosine function to find the measures of angles A, B, and C.
A = cos^(-1)(9 / 16) ≈ 39.23°
B = cos^(-1)(1 / 8) ≈ 82.88°
C = cos^(-1)(3 / 4) ≈ 41.41°
Therefore, the angles of triangle ABC in order from smallest to largest are:
A ≈ 39.23°, C ≈ 41.41°, B ≈ 82.88°