In the triangle the length of side a is 5 ft and the measurement of angle A = 60 degrees. Find the exact lengths of sides b and c.

Question

GPT 3.5 · Answer

To find the lengths of sides b and c, we can use the Law of Sines.

The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of the opposite angle is constant. In equation form, this is given by:

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

In this case, we know the length of side a is 5 ft and the measurement of angle A is 60 degrees.

So, we have:

5 ft / sin(60 degrees) = b / sin(B) = c / sin(C)

To find the length of side b, we can rearrange the equation to solve for b:

b = (5 ft / sin(60 degrees)) * sin(B)

To find the length of side c, we can rearrange the equation to solve for c:

c = (5 ft / sin(60 degrees)) * sin(C)

Now, we need to find the measures of angles B and C. In a triangle, the sum of the measures of the three angles is 180 degrees.

So, we have:

angle B + angle C = 180 degrees

Since we know angle A is 60 degrees, we can substitute it into the equation:

60 degrees + angle B + angle C = 180 degrees

Simplifying, we get:

angle B + angle C = 120 degrees

Since angles B and C are complementary to angle A, we know that:

angle B = 180 degrees - 60 degrees = 120 degrees

angle C = 180 degrees - 60 degrees = 120 degrees

Now, we can substitute the values into the equations for b and c:

b = (5 ft / sin(60 degrees)) * sin(120 degrees)

c = (5 ft / sin(60 degrees)) * sin(120 degrees)

Using a calculator or trigonometric table, we can find that sin(120 degrees) = √3 / 2.

So, we have:

b = (5 ft / sin(60 degrees)) * (√3 / 2)

c = (5 ft / sin(60 degrees)) * (√3 / 2)

Simplifying further, we get:

b = (5 ft * √3) / (2 * sin(60 degrees))

c = (5 ft * √3) / (2 * sin(60 degrees))

Using a calculator or trigonometric table, we can find that sin(60 degrees) = √3 / 2.

Substituting the value into the equations:

b = (5 ft * √3) / (2 * ( √3 / 2))

c = (5 ft * √3) / (2 * ( √3 / 2))

Simplifying:

b = 5 ft

c = 5 ft

Therefore, the exact lengths of sides b and c are 5 ft each.