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)

Answer 1

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