Question

Given that angle B is ∏ /3 and side b is 8 feet, determine the lengths of sides a and c. Do not round answers to decimals.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine the lengths of sides a and c, we will use the Law of Sines.

The Law of Sines states that for any triangle with sides a, b, c and opposite angles A, B, C, the following relationship holds:

a / sin(A) = b / sin(B) = c / sin(C)

In this case, we are given angle B = π/3 and side b = 8 feet. Let's solve for side a first.

Since the sum of the angles in a triangle is always π (180 degrees), angle A = π - B - C.
Since the sum of the angles opposite to their respective sides in a triangle is also always π (180 degrees), angle A is opposite to side a.

So, angle A = π - B - C = π - π / 3 - C = 2π / 3 - C

Using the Law of Sines, we have:

a / sin(A) = b / sin(B)

a / sin(2π / 3 - C) = 8 / sin(π / 3)

Now, we need to determine angle C.

The sum of the angles in a triangle is always π (180 degrees), so angle C = π - A - B = π - 2π / 3 - π / 3 = π / 3

Substituting angle C = π / 3 into the equation:

a / sin(2π / 3 - π / 3) = 8 / sin(π / 3)

a / sin(π / 3) = 8 / sin(π / 3)

a = 8

So, side a is 8 feet.

Now, let's solve for side c.

Using the Law of Sines, we have:

c / sin(C) = b / sin(B)

c / sin(π / 3) = 8 / sin(π / 3)

c = 8

So, side c is also 8 feet.