SOLUTION: If the arc of a doorway arch is 60cm high and the width is 200cm what would be the circumstance of the circle that would contain that arch ?

Question 1081136: If the arc of a doorway arch is 60cm high and the width is 200cm what would be the circumstance of the circle that would contain that arch ?
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
The arc sits at the top of the circle, and the width is a horizontal chord from one side of the circle to the other.
For simplicity, let the center of the circle be at the origin (0,0).
Let h = the distance from the center of the circle to the midpoint of the arch.
Then the coordinates for the arch points which lie on the circle are (-100,h) (100,h) and (0,h+60)
From the definition of the circle, the distance from the center to each of the points is a constant, equal to the radius.
Therefore we can write d1 = sqrt(100^2+h^2) = d2 = sqrt(h+60)^2) = h+60
Squaring both sides and simplifying we get
h^2 + 120h + 3600 = 10000 + h^2
12h = 640 -> h = 53 1/3
Thus the radius of the circle = 53 1/3 + 60 = 113 1/3
The circumference is 2*pi*r = 712.1 cm

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