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°
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)
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