Math Problem Statement

Circle passes through (2,-4) and has its center at (5,7). Find the radius

Solution

To find the radius of the circle, we can use the distance formula between the center of the circle and any point on the circle. The general formula for the distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In this case:

  • The center of the circle is (5,7)(5, 7)
  • A point on the circle is (2,4)(2, -4)

Now, substitute these coordinates into the distance formula:

d=(25)2+(47)2d = \sqrt{(2 - 5)^2 + (-4 - 7)^2}

d=(3)2+(11)2d = \sqrt{(-3)^2 + (-11)^2}

d=9+121d = \sqrt{9 + 121}

d=130d = \sqrt{130}

Therefore, the radius rr of the circle is:

r=13011.4r = \sqrt{130} \approx 11.4

Thus, the radius of the circle is approximately 11.411.4 units.


Would you like more details on how to approach this or have any other questions? Here are some related questions you could explore:

  1. How does the distance formula apply in three-dimensional space?
  2. What is the general equation of a circle when the center and radius are known?
  3. How do you calculate the area of a circle using the radius?
  4. What happens if the point lies outside the circle? How does it affect the radius?
  5. How do you find the equation of a circle if you know two points and the center?

Tip: Remember that the distance formula is just an application of the Pythagorean theorem in coordinate geometry!

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Circles
Distance Formula

Formulas

Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Pythagorean Theorem (used indirectly in the distance formula)

Suitable Grade Level

Grades 8-10