A circle passes through the points (1, 2), (3, 4), and (5, 6). What is the radiu

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Question: A circle passes through the points (1, 2), (3, 4), and (5, 6). What is the radius of the circle?

Options:

Correct Answer: 3

Solution:

Using the distance formula, the radius can be calculated from the center found using the circumcircle method.

A circle passes through the points (1, 2), (3, 4), and (5, 6). What is the radius of the circle?

A circle passes through the points (1, 2), (3, 4), and (5, 6). What is the radius of the circle?

Step 1: Identify the points through which the circle passes. The points are (1, 2), (3, 4), and (5, 6).
Step 2: Use the circumcircle method to find the center of the circle. This involves finding the perpendicular bisectors of the segments formed by the points.
Step 3: Calculate the midpoints of the segments between the points. For example, the midpoint between (1, 2) and (3, 4) is ((1+3)/2, (2+4)/2) = (2, 3).
Step 4: Find the slope of the line segment between the points. For the segment from (1, 2) to (3, 4), the slope is (4-2)/(3-1) = 1.
Step 5: Determine the slope of the perpendicular bisector. If the slope of the segment is 1, the slope of the perpendicular bisector is -1.
Step 6: Write the equation of the perpendicular bisector using the midpoint and the slope. For the midpoint (2, 3) and slope -1, the equation is y - 3 = -1(x - 2).
Step 7: Repeat steps 3 to 6 for another pair of points, such as (3, 4) and (5, 6), to find another perpendicular bisector.
Step 8: Solve the two equations of the perpendicular bisectors simultaneously to find the center of the circle (h, k).
Step 9: Use the distance formula to calculate the radius. The radius is the distance from the center (h, k) to any of the points, for example, (1, 2).
Step 10: The distance formula is r = sqrt((h - 1)^2 + (k - 2)^2). Calculate this to find the radius.

Circle Geometry – Understanding the properties of circles, including how to find the center and radius using points on the circumference.
Distance Formula – Applying the distance formula to calculate distances between points, which is essential for determining the radius.
Circumcircle Method – Using the circumcircle method to find the center of a circle that passes through multiple points.