Math Problem Statement
Solution
Let's solve the problem step by step based on the image:
We are given two endpoints of a diameter of a circle: and .
a.) Find the center of the circle.
The center of a circle that has its diameter defined by two points is the midpoint of those two points. The midpoint between two points and is calculated using:
Using the endpoints and :
So, the center of the circle is .
b.) Determine the radius of the circle.
The radius is half the length of the diameter. We can find the length of the diameter by calculating the distance between the two points and using the distance formula:
Substituting the values:
Since the diameter is 8 units, the radius is:
c.) Write the equation of the circle.
The standard form of a circle's equation is: where is the center and is the radius.
From parts (a) and (b), the center is and the radius is . Plugging these into the equation:
which simplifies to:
This is the equation of the circle.
Would you like any more details on these steps?
Here are five related questions to expand on this:
- How do you find the center if only one point and the radius are given?
- Can a circle have a diameter along a non-vertical or non-horizontal line?
- How do you find the equation of a circle given its center and circumference?
- What is the relationship between the diameter and circumference of a circle?
- How can you find the area of a circle from its equation?
Tip: For problems involving the distance between two points, always recall the distance formula.
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Math Problem Analysis
Mathematical Concepts
Geometry
Circle Equations
Coordinate Geometry
Formulas
Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Standard equation of a circle: (x - h)^2 + (y - k)^2 = r^2
Theorems
Midpoint Theorem
Distance Formula
Suitable Grade Level
Grades 9-10
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