Math Problem Statement

Sresha is trying to find the equation of a circle. She knows that two endpoints of a diameter are (-2, -1) and (-2, 7). a) Find the center of the circle. b) Determine the radius of the circle. c) Write the equation of the circle.

Solution

Let's solve the problem step by step based on the image:

We are given two endpoints of a diameter of a circle: (2,1)(-2, -1) and (2,7)(-2, 7).

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 MM between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is calculated using:

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Using the endpoints (2,1)(-2, -1) and (2,7)(-2, 7): M=(2+(2)2,1+72)=(2,3)M = \left( \frac{-2 + (-2)}{2}, \frac{-1 + 7}{2} \right) = (-2, 3)

So, the center of the circle is (2,3)(-2, 3).

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 (2,1)(-2, -1) and (2,7)(-2, 7) using the distance formula:

d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Substituting the values: d=(2(2))2+(7(1))2=(0)2+(8)2=8d = \sqrt{(-2 - (-2))^2 + (7 - (-1))^2} = \sqrt{(0)^2 + (8)^2} = 8

Since the diameter is 8 units, the radius is: Radius=82=4\text{Radius} = \frac{8}{2} = 4

c.) Write the equation of the circle.

The standard form of a circle's equation is: (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2 where (h,k)(h, k) is the center and rr is the radius.

From parts (a) and (b), the center is (2,3)(-2, 3) and the radius is 44. Plugging these into the equation:

(x(2))2+(y3)2=42(x - (-2))^2 + (y - 3)^2 = 4^2 which simplifies to: (x+2)2+(y3)2=16(x + 2)^2 + (y - 3)^2 = 16

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:

  1. How do you find the center if only one point and the radius are given?
  2. Can a circle have a diameter along a non-vertical or non-horizontal line?
  3. How do you find the equation of a circle given its center and circumference?
  4. What is the relationship between the diameter and circumference of a circle?
  5. 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