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Difficulty: EasyDistance and Midpoint Formulas

A circle in the standard (x,y)(x,y) coordinate plane has its center at (2,3)(2, 3) and passes through the point (8,11)(8, 11). What is the length of the radius of this circle?

  1. 10Answer
  2. B
    14
  3. C
    28
  4. D
    70
  5. E
    100

Answer

10
The correct answer is 10. The radius of the circle is the distance from the center (2,3)(2,3) to the point (8,11)(8,11) on the circle. Substituting these coordinates into the distance formula yields (8−2)2+(11−3)2=62+82=36+64=100=10\sqrt{(8-2)^2 + (11-3)^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10.

Step-by-Step Solution

1
Identify the formula for the radius of a circle given its center and a point on the circle.
The radius rr is the distance between the center (x1,y1)=(2,3)(x_1, y_1) = (2, 3) and the point (x2,y2)=(8,11)(x_2, y_2) = (8, 11).
By definition, the radius of a circle is the straight-line distance from the center to any point on its outer boundary.
2
Substitute the coordinates into the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
r=(8−2)2+(11−3)2r = \sqrt{(8 - 2)^2 + (11 - 3)^2}
This sets up the calculation for the horizontal and vertical changes between the center and the point on the circle.
3
Simplify the terms inside the parentheses and evaluate the exponents.
r=62+82=36+64r = \sqrt{6^2 + 8^2} = \sqrt{36 + 64}
Following the order of operations, we first perform the subtraction within the parentheses, and then evaluate the squares.
4
Add the values inside the radical and take the square root.
r=100=10r = \sqrt{100} = 10
We must sum the terms under the square root before taking the square root of the total sum.

Key Concept

Using the distance formula to find the radius of a circle given its center and a point on the circle.
Estimated Time:45s
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Topics

Distance and Midpoint Formulas

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