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Chapter 4: Problem 5

A diameter of a circle has endpoints with coordinates \((2,6)\) and \((-4,10) .\) Find the coordinates of the center of the circle.

Short Answer

Expert verified
The coordinates of the center of the circle are (-1, 8).

Step by step solution

01

Identify the Coordinates of the Endpoints

First, identify the coordinates of the endpoints of the diameter. In this problem the endpoints are A = (2,6) and B = (-4,10).
02

Use the Midpoint Formula

To find the coordinates of the midpoint, use the midpoint formula which is ((x1+x2)/2 , (y1+y2)/2) where (x1, y1) and (x2, y2) are the coordinates of the endpoints. Here, x1 = 2, y1 = 6, x2 = -4 and y2 = 10.
03

Calculate the Midpoint

Substitute the values into the midpoint formula to find the coordinates of the midpoint. The midpoint will be ((2+(-4))/2 , (6+10)/2 ) which simplifies to (-1,8).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Geometry
Circle geometry is a fascinating area of mathematics that deals with the properties and relations of points, lines, angles, and figures in a circle. A circle is defined as the set of all points that are equidistant from a fixed center point in a plane. The distance from the center point to any point on the circle is called the radius.
  • The diameter is one of the special line segments in a circle. It passes through the center and touches two points on the circle's circumference. Consequently, the diameter is always twice the length of the radius.

  • In a circle, any line segment passing through the center and having its endpoints on the circle is a diameter. Thus, all diameters of a circle are equal in length.
Understanding these basic properties helps you solve various problems related to circle geometry. For example, knowing that the midpoint of a diameter is also the center can be very helpful.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, combines algebra and geometry using a coordinate plane. This allows you to find geometric properties and solve geometric problems algebraically, involving points, lines, and curves.
  • It uses a system of coordinates to describe the position of points. Typically, a point in this system is denoted as \( (x, y) \), where \( x \) represents the horizontal position and \( y \) the vertical position.

  • The key tools of coordinate geometry include distance formula, midpoint formula, and the slope of a line, among others.
Using these concepts makes problems more straightforward by translating geometric shapes into numeric equations quicker and solving them efficiently. This is particularly useful when working with complex shapes or configurations in geometry.
Diameter of a Circle
The diameter of a circle is a straight line segment that joins two points on the circle and passes through the center. Because the diameter divides the circle into two equal parts, each half is known as a semicircle.
  • The formula for diameter, if given a radius, can be expressed as \( d = 2r \), where \( d \) is the diameter and \( r \) is the radius.

  • When you know the endpoints of a diameter, you can find the center of the circle using the midpoint formula: \( \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \). This finds the average of the x-coordinates and y-coordinates, effectively giving you the center point.
Using the midpoint formula to find the center is particularly helpful when dealing with geometric problems on a coordinate plane. It solves problems not only related to circle centers but also helps with finding the balance or symmetry in various shapes.

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Most popular questions from this chapter

To the nearest second, what is the first time after 7: 00 that the hands of a clock form a right angle?

Prove: The angle bisector of the vertex angle of an isosceles triangle is perpendicular to the base.

A four-sided figure with two disjoint pairs of consecutive sides congruent is called a kite. The two segments joining opposite vertices are its diagonals. Prove that one of these diagonals is the perpendicular bisector of the other diagonal.

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draw your own diagram and write "Given:" and "Prove:" statements in terms of your diagram. Given: An isosceles triangle and the median to the base Prove: The median is the perpendicular bisector of the base. (This sentence contains two conclusions - "the median is perpendicular to the base" and "the median bisects the base."
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