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Difficulty: MediumDistance and Midpoint Formulas

In the standard (x,y)(x, y) coordinate plane, the center of a circle is located at the midpoint of the line segment with endpoints A(12,9)A\left(\frac{1}{2}, 9\right) and B(92,3)B\left(\frac{9}{2}, 3\right). If the circle passes through the point P(132,9)P\left(\frac{13}{2}, 9\right), what is the radius of the circle?

  1. A
    6
  2. B
    7
  3. 5Answer
  4. D
    25
  5. E
    2132\sqrt{13}

Answer

The radius of the circle is 5.
The center of the circle is the midpoint of segment AB. Calculating the midpoint of A(0.5,9)A(0.5, 9) and B(4.5,3)B(4.5, 3) gives M(2.5,6)M(2.5, 6). The radius is the distance from this center M(2.5,6)M(2.5, 6) to the given point P(6.5,9)P(6.5, 9) on the circle. The distance is (6.5−2.5)2+(9−6)2=42+32=25=5\sqrt{(6.5 - 2.5)^2 + (9 - 6)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.

Step-by-Step Solution

1
Find the coordinates of the center of the circle by calculating the midpoint of segment AB.
Center M=(1/2+9/22,9+32)=(52,6)=(2.5,6)M = \left(\frac{1/2 + 9/2}{2}, \frac{9 + 3}{2}\right) = \left(\frac{5}{2}, 6\right) = (2.5, 6).
The center of the circle is located at the midpoint of the line segment AB.
2
Use the distance formula to find the distance between the center M(2.5,6)M(2.5, 6) and the point P(6.5,9)P(6.5, 9) on the circle.
r=(6.5−2.5)2+(9−6)2=42+32=25=5r = \sqrt{(6.5 - 2.5)^2 + (9 - 6)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.
The radius of a circle is the distance from its center to any point on its boundary.

Key Concept

Using the midpoint formula to locate the center of a circle, and then applying the distance formula between the center and a boundary point to find the radius.

Alternative Method

You can visualize the horizontal and vertical distances on a coordinate plane. The horizontal distance between the x-coordinates of the center (2.5,6)(2.5, 6) and the point (6.5,9)(6.5, 9) is 6.5−2.5=46.5 - 2.5 = 4. The vertical distance is 9−6=39 - 6 = 3. Recognizing the 3-4-5 right triangle triplet immediately gives the distance (hypotenuse) as 5.
Estimated Time:1m 15s
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Topics

Distance and Midpoint Formulas

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