Sender: Howard B Davis Subject: GEOMETRIC PROBLEM This is a problem inÊreal life, but I'm just trying to find the solution for my own satisfaction although it has practical application. We start a Line that goes up 1 unit, then it goes to the Right for 5 units long, and then goes down 1 unit which is the end point. If we draw a circle that is tangent to both ends as well as the midpoint of the horizontal line: How do we find the radius of the arc; in Mathematics, with only this information? So, if anyone knowsÊhow to get the answer. I willÊreallyÊappreciate the information. THANKS. Hi Howard,The situation the way you describe it is impossible. At any point P on a circle the tangent line at P is perpendicular to the line joining P to the center ( a radius). For your situation that would mean that the center is at C in the diagram which it clearly can't be. If we draw a circle is only required to pass through both ends as well as the midpoint of the horizontal line then there is a solution that can be described by putting a coordinate system on the plane.
In the diagram C, the center of the circle, has coordinates (0,a). Let r be the radius of the circle then r = CP = a + 1. CQ is also a radius of the circle and triangle CQO is a right triangle. Thus by the theorem of Pythagoras, Thus a^{2} + 2 a + 1 = 2.5^{2} + a^{2} 2 a + 2 = 2.5^{2} + 1 2(a + 1) = 6.25 + 1 r = a + 1 = 7.25/2 = 3 5/8
Cheers,
