The formula used to determine the radius of the yaw mark arc is derived from a geometric relationship about two intersecting chords in a circle. In the figure, chords AB and CD intersect at point E in the circle. The product of the two segment lengths making up chord AB, AE × EB, is equal to the product of the two segment lengths making up chord CD, CE × ED.
In the next figure, the yaw mark is darkened and it is continued to form a complete circle. A chord is drawn connecting two points on the yaw mark. The middle ordinate is also drawn. The length of the middle ordinate is M and the length of the chord is CD. The middle ordinate cuts the chord into two equal pieces with each half of the chord CD/2 units in length. The radius of the circle has length r as shown in the diagram. Applying the property to the two intersecting chords in this diagram, you get AE × EB = CE × ED.

a. From the diagram, CE = CD/2, ED = CD/2, and EB = M. You need to determine the length of the segment AE. Notice that AB = 2r. (It is a diameter, which equals the length of two radii.) Also notice that AE = AB ˆ’ EB. Write an algebraic expression that represents the length of AE.
b. Write the algebraic expression for the product of the segments of a chord that applies to this situation. Do not simplify.
c. Simplify the side of the equation that represents the product of the segments of chord CD. Write the new equation.
d. Solve the equation for r by isolating the variable r on one side of the equation. Show your work. Compare your answer with the radius formula.

Transcribed Image Text:

Center YAW MARK