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)
To find the angles of triangle ABC, we need to use the fact that the sum of the angles in a triangle is always 180 degrees.
Let's denote the angles of triangle ABC as ∠A, ∠B, and ∠C.
From the given information, we know that the perimeter of triangle ABC is 30 ft. Therefore, we can write the equation:
AB + BC + AC = 30
Since AB = 3x+1, BC = 4x, and AC = 5x-7, we can substitute these values into the equation:
(3x+1) + (4x) + (5x-7) = 30
Simplifying the equation:
12x - 6 = 30
Adding 6 to both sides:
12x = 36
Dividing both sides by 12:
x = 3
Now that we have found the value of x, we can substitute it back into the lengths of the sides to find their values:
AB = 3x+1 = 3(3)+1 = 10 ft
BC = 4x = 4(3) = 12 ft
AC = 5x-7 = 5(3)-7 = 8 ft
Now, we can use the lengths of the sides to find the angles of triangle ABC.
To find ∠A, we can use the Law of Cosines:
cos(∠A) = (BC² + AC² - AB²) / (2 * BC * AC)
cos(∠A) = (12² + 8² - 10²) / (2 * 12 * 8)
cos(∠A) = (144 + 64 - 100) / (2 * 12 * 8)
cos(∠A) = 108 / 192
cos(∠A) = 9 / 16
Using a calculator or trigonometric tables, we can find the value of ∠A:
∠A ≈ 48.59 degrees
To find ∠B, we can use the Law of Cosines again:
cos(∠B) = (AC² + AB² - BC²) / (2 * AC * AB)
cos(∠B) = (8² + 10² - 12²) / (2 * 8 * 10)
cos(∠B) = (64 + 100 - 144) / (2 * 8 * 10)
cos(∠B) = 20 / 160
cos(∠B) = 1 / 8
Again, using a calculator or trigonometric tables, we can find the value of ∠B:
∠B ≈ 82.87 degrees
Now, we can find ∠C by subtracting the sum of ∠A and ∠B from 180 degrees:
∠C = 180 - ∠A - ∠B
∠C = 180 - 48.59 - 82.87
∠C ≈ 48.54 degrees
Therefore, the angles of triangle ABC in order from smallest to largest are approximately:
∠A ≈ 48.59 degrees
∠C ≈ 48.54 degrees
∠B ≈ 82.87 degrees