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Difficulty: EasyEquations and Graphs of Circles

A circle drawn in the standard (x,y)(x, y) coordinate plane has its center at the point (3,−2)(3, -2) and passes through the point (7,1)(7, 1). What is the radius of this circle?

Answer: 5

Answer

The radius of the circle is 5.
The radius of a circle is defined as the distance from its center to any point on the boundary. Using the distance formula between the center (3,−2)(3, -2) and the point (7,1)(7, 1) yields r=(7−3)2+(1−(−2))2=42+32=25=5r = \sqrt{(7 - 3)^2 + (1 - (-2))^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.

Step-by-Step Solution

1
Identify the formula to find the distance between two points, which represents the radius of the circle.
The distance formula is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
The radius of a circle is the distance from its center to any point on its circumference.
2
Substitute the coordinates of the center (3,−2)(3, -2) and the point (7,1)(7, 1) into the formula.
r=(7−3)2+(1−(−2))2r = \sqrt{(7 - 3)^2 + (1 - (-2))^2}
We substitute x1=3,y1=−2x_1 = 3, y_1 = -2 and x2=7,y2=1x_2 = 7, y_2 = 1 into the distance formula.
3
Simplify the expression to determine the radius.
r=42+32=16+9=25=5r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
Calculate the squared differences, add them, and take the square root to find the final value.

Key Concept

Calculating the radius of a circle given its center and a point on the circle using the distance formula.
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Topics

Equations and Graphs of Circles

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