SOLUTION: please help me! a piece of a broken circular gear is brought into a metal shop so that a replacement can be built. a ruler is placed across two points on the rim, and the len

Question 149127: please help me!

a piece of a broken circular gear is brought into a metal shop so that a replacement can be built. a ruler is placed across two points on the rim, and the length of the chordd is found to be 14 inches. the distance from the midpoint of this chord to the nearest point on the rim is found to be 4 inches. find the radius of the original gear.
Answer by scott8148(6628) (Show Source): You can put this solution on YOUR website!
a diameter of the circle passes through the midpoint of the chord
__ this diameter is the hypotenuse of a right triangle inscribed in half of the circle

the altitude of this triangle (from the diameter) is 7 (half the chord length)
__ the distance from one end of the diameter to the altitude is 4

the altitude divides the inscribed right triangle into two smaller right triangles that are similar to the large triangle

let x="diameter of circle"

because the triangles are similar __ 4:7::7:(x-4) __ or 4/7=7/(x-4) __ "cross" multiplying __ 4x-16=49

adding 16 __ 4x=65 __ dividing by 4 __ x=16.25

radius is half of the diameter so radius is 8.125
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