Question

Round your answer to the nearest hundredth. A circle with center P and three points, A, B, and C, on the circle. Point P is on segment C B. Segment P A is drawn such that angle A P B measures 45 degrees. The length of segment C B is 8 feet. The arc length is about feet.

Answer 1

Explanation:

Since segment P A bisects angle A P B, we know that angle A P C measures 90 degrees. Therefore, segment C P is the radius of the circle.

We can use the Pythagorean theorem to find the length of segment A P:

AP^2 + CP^2 = AC^2

AP^2 + CP^2 = (2CP)^2 (since AC is the diameter of the circle)

AP^2 = 4CP^2 - CP^2 = 3CP^2

AP = CP * sqrt(3)

We know that CP is 4 feet, so:

AP = 4 * sqrt(3) feet

The arc length of a circle is given by:

arc length = (angle/360) * 2 * pi * radius

The angle of arc A B C is 360 - 45 = 315 degrees. The radius of the circle is CP = 4 feet. Substituting these values into the formula, we get:

arc length = (315/360) * 2 * pi * 4 feet

arc length = 3.5 * pi feet

Rounding to the nearest hundredth, the arc length is approximately 10.99 feet.