The perimeter of ABC is 30 feet. AB=5x-7, BC=3x+1, AC=4x. List the angles of ABC in order from smallest to largest
A) Angle A, B, C
B) Angle A, C, B
C) Angle C, B, A
D) Angle C, A, B
To find the angles of a triangle, we must use the fact that the sum of the angles in a triangle is 180 degrees.
The perimeter of a triangle is the sum of the lengths of its sides. In this case, we have AB + BC + AC = 5x - 7 + 3x + 1 + 4x = 12x - 6.
Given that the perimeter is 30 feet, we can set up the equation 12x - 6 = 30 and solve for x: 12x = 36, x = 3.
Now we can substitute the value of x back into the expressions for the lengths of the sides: AB = 5(3) - 7 = 8, BC = 3(3) + 1 = 10, AC = 4(3) = 12.
To find the angles, we can use the Law of Cosines:
AC² = AB² + BC² - 2(AB)(BC)cos(A)
12² = 8² + 10² - 2(8)(10)cos(A)
144 = 64 + 100 - 160cos(A)
144 = 164 - 160cos(A)
-20 = -160cos(A)
cos(A) = -20/-160
cos(A) = 1/8
A = arccos(1/8)
A ≈ 82.82°
AB² = AC² + BC² - 2(AC)(BC)cos(B)
8² = 12² + 10² - 2(12)(10)cos(B)
64 = 144 + 100 - 240cos(B)
64 = 244 - 240cos(B)
-180 = -240cos(B)
cos(B) = -180/-240
cos(B) = 3/4
B = arccos(3/4)
B ≈ 41.41°
BC² = AB² + AC² - 2(AB)(AC)cos(C)
10² = 8² + 12² - 2(8)(12)cos(C)
100 = 64 + 144 - 192cos(C)
100 = 208 - 192cos(C)
-108 = -192cos(C)
cos(C) = -108/-192
cos(C) = 27/48
C = arccos(27/48)
C ≈ 56.77°
The angles of ABC, listed in order from smallest to largest, are B, A, C.
Therefore, the correct answer is:
D) Angle C, A, B