SOLUTION: The center of a circle is at (4,2) and its radius is 5.Find the length of the chord which is bisected at (2,-1)

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Question 1016836: The center of a circle is at (4,2) and its radius is 5.Find the length of the chord which is bisected at (2,-1)
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39621) About Me  (Show Source):
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Line containing (4,2) and (2,-1) will have slope -3%2F2.

The bisected chord will be on the line of equation y-%28-1%29=%282%2F3%29%28x-2%29,
y=%282%2F3%29x-7%2F3.

The points on the circle of the bisected chord will the the intersection points of y=%282%2F3%29x-7%2F3 and %28x-4%29%5E2%2B%28y-2%29%5E2=25. Use the distance formula to find the distance between these two points.

Answer by ikleyn(52824) About Me  (Show Source):
You can put this solution on YOUR website!
.
The center of a circle is at (4,2) and its radius is 5.Find the length of the chord which is bisected at (2,-1)
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1. Make a sketch.

   Draw the circle, the chord and the bisection point.
   Draw the radii of the circle to the endpoints of the chord.
   Draw the line from the center of the circle to the bisection point at the chord.
   This line is the perpendicular to the chord.
   Do you understand why? - Because it is the median in the isosceles triangle.

   Therefore, the plan of the solution is:
      a) find the length of this perpendicular, and then
      b) find the length of the leg of the right-angled triangle, which is exactly half of the length of the chord; and then
      c) double the length of that leg.


2. Below find how this plan is implemented.

a) The distance from the center (4,2) to the point (2,-1) is

   sqrt%28%284-2%29%5E2+%2B+%282-%28-1%29%29%5E2%29 = sqrt%282%5E2+%2B+3%5E2%29 = sqrt%284%2B9%29 = sqrt%2813%29.


b) The length of the leg of the right-angled triangle is
   sqrt%285%5E2+-+13%29 = sqrt%2825-13%29 = sqrt%2812%29 = 2%2Asqrt%283%29.

c) The length of the chord is 2%2A2%2Asqrt%283%29 = 4%2Asqrt%283%29.

Answer. The length of the chord is 4%2Asqrt%283%29.