Question

Find the measure and length of arc ABC. Express the answers in terms of pi.

Answer 1

The measure of arc ABC is 46°.

The length of arc ABC is approximately 3.23, or 3.23π.

Finding the measure of arc ABC:

First, we need to find the measure of central angle ∠AOB. We are given that ∠BOC = 83° and ∠AOC = 51°. Since the sum of the measures of the angles in a triangle is 180°, we can find ∠AOB as follows:

• ∠AOB = 180° - ∠BOC - ∠AOC
• ∠AOB = 180° - 83° - 51°
• ∠AOB = 46°

Therefore, the measure of arc ABC is also 46°.

Finding the length of arc ABC:

• We are given that the radius of the circle is 4.
• The formula for the length of an arc is s = r * θ, where s is the arc length, r is the radius, and θ is the central angle in radians.

First, we need to convert the central angle from degrees to radians. We know that there are 2π radians in a full circle, so we can use the following conversion factor:

• 1 radian = 180° / π

Therefore, the measure of central angle ∠AOB in radians is:

• θ = 46° * (π / 180°)
• θ = 0.807 radians

Now we can find the length of arc ABC:

• s = 4 * 0.807
• s = 3.228

Therefore, the length of arc ABC is approximately 3.23, which can also be expressed as 3.23π.