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Difficulty: MediumDistance and Midpoint Formulas

In the standard (x,y)(x, y) coordinate plane, the midpoint of a line segment with endpoints (1,−2)(1, -2) and (7,10)(7, 10) is the center of a circle. If the point (8,7)(8, 7) lies on the circle, what is the radius of the circle?

  1. A
    7
  2. B
    25
  3. C
    353\sqrt{5}
  4. 5Answer
  5. E
    26\sqrt{26}

Answer

The radius of the circle is 5.
The midpoint of the line segment with endpoints (1,−2)(1, -2) and (7,10)(7, 10) is calculated as (1+72,−2+102)=(4,4)(\frac{1+7}{2}, \frac{-2+10}{2}) = (4, 4), which is the center of the circle. The distance from the center (4,4)(4, 4) to the point (8,7)(8, 7) on the circle is the radius, which is (8−4)2+(7−4)2=16+9=5\sqrt{(8-4)^2 + (7-4)^2} = \sqrt{16+9} = 5.

Step-by-Step Solution

1
Find the midpoint of the line segment with endpoints (1,−2)(1, -2) and (7,10)(7, 10) to determine the center of the circle.
The center of the circle is (4,4)(4, 4).
The midpoint of a segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by (x1+x22,y1+y22)(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}). Calculating this gives (1+72,−2+102)=(4,4)(\frac{1+7}{2}, \frac{-2+10}{2}) = (4, 4).
2
Calculate the distance between the center of the circle (4,4)(4, 4) and the point (8,7)(8, 7) on the circle to find the radius.
The radius of the circle is 5.
The distance formula is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}. Substituting the coordinates gives (8−4)2+(7−4)2=42+32=16+9=25=5\sqrt{(8-4)^2 + (7-4)^2} = \sqrt{4^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5.

Key Concept

Finding the midpoint of a line segment to determine a circle's center and using the distance formula to calculate its radius.
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Topics

Distance and Midpoint Formulas

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