Distance and Midpoint Formulas; Circles – Intermediate Algebra (2024)

Learning Objectives

By the end of this section, you will be able to:

• Use the Distance Formula
• Use the Midpoint Formula
• Write the equation of a circle in standard form
• Graph a circle

Before you get started, take this readiness quiz.

1. Find the length of the hypotenuse of a right triangle whose legs are 12 and 16 inches.
   If you missed this problem, review (Figure).
2. Factor:
   If you missed this problem, review (Figure).
3. Solve by completing the square:
   If you missed this problem, review (Figure).

Use the Distance Formula

We have used the Pythagorean Theorem to find the lengths of the sides of a right triangle. Here we will use this theorem again to find distances on the rectangular coordinate system. By finding distance on the rectangular coordinate system, we can make a connection between the geometry of a conic and algebra—which opens up a world of opportunities for application.

Our first step is to develop a formula to find distances between points on the rectangular coordinate system. We will plot the points and create a right triangle much as we did when we found slope in Graphs and Functions. We then take it one step further and use the Pythagorean Theorem to find the length of the hypotenuse of the triangle—which is the distance between the points.

Use the rectangular coordinate system to find the distance between the points and

Plot the two points. Connect the two points with a line. Draw a right triangle as if you were going to find slope.
Find the length of each leg.
Use the Pythagorean Theorem to find d, the distance between the two points.
Substitute in the values.
Simplify.
Use the Square Root Property.
Since distance, d is positive, we can eliminate	The distance between the points and is 5.

The method we used in the last example leads us to the formula to find the distance between the two points and

When we found the length of the horizontal leg we subtracted which is

When we found the length of the vertical leg we subtracted which is

If the triangle had been in a different position, we may have subtracted or The expressions and vary only in the sign of the resulting number. To get the positive value-since distance is positive- we can use absolute value. So to generalize we will say and

In the Pythagorean Theorem, we substitute the general expressions and rather than the numbers.

This is the Distance Formula we use to find the distance d between the two points and

Use the Midpoint Formula

It is often useful to be able to find the midpoint of a segment. For example, if you have the endpoints of the diameter of a circle, you may want to find the center of the circle which is the midpoint of the diameter. To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Use the Midpoint Formula to find the midpoint of the line segments whose endpoints are and Plot the endpoints and the midpoint on a rectangular coordinate system.

Write the Midpoint Formula.
Label the points, and substitute.
Simplify.
	The midpoint of the segment is the point
Plot the endpoints and midpoint.

Both the Distance Formula and the Midpoint Formula depend on two points, and It is easy to confuse which formula requires addition and which subtraction of the coordinates. If we remember where the formulas come from, is may be easier to remember the formulas.

Write the Equation of a Circle in Standard Form

As we mentioned, our goal is to connect the geometry of a conic with algebra. By using the coordinate plane, we are able to do this easily.

We define a circle as all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, and the fixed distance is called the radius, r, of the circle.

We look at a circle in the rectangular coordinate system. The radius is the distance from the center, to a point on the circle,
To derive the equation of a circle, we can use the distance formula with the points and the distance, r.
Substitute the values.
Square both sides.

This is the standard form of the equation of a circle with center, and radius, r.

Write the standard form of the equation of the circle with radius 3 and center

Use the standard form of the equation of a circle
Substitute in the values and
Simplify.

In the last example, the center was Notice what happened to the equation. Whenever the center is the standard form becomes

Write the standard form of the equation of the circle with radius 2 and center

Use the standard form of the equation of a circle.
Substitute in the values.
Simplify.

In the next example, the radius is not given. To calculate the radius, we use the Distance Formula with the two given points.

Graph a Circle

Any equation of the form is the standard form of the equation of a circle with center, and radius, r. We can then graph the circle on a rectangular coordinate system.

Note that the standard form calls for subtraction from x and y. In the next example, the equation has so we need to rewrite the addition as subtraction of a negative.

Find the center and radius, then graph the circle:

Use the standard form of the equation of a circle. Identify the center, and radius, r.
	Center: radius: 3
Graph the circle.

To find the center and radius, we must write the equation in standard form. In the next example, we must first get the coefficient of to be one.

Find the center and radius and then graph the circle,

Divide each side by 4.
Use the standard form of the equation of a circle. Identify the center, and radius, r.
	Center: radius: 4
Graph the circle.

If we expand the equation from (Figure), the equation of the circle looks very different.

This form of the equation is called the general form of the equation of the circle.

If we are given an equation in general form, we can change it to standard form by completing the squares in both x and y. Then we can graph the circle using its center and radius.

ⓐ Find the center and radius, then ⓑ graph the circle:

We need to rewrite this general form into standard form in order to find the center and radius.

Group the x-terms and y-terms. Collect the constants on the right side.
Complete the squares.
Rewrite as binomial squares.
Identify the center and radius.	Center: radius: 3
Graph the circle.

In the next example, there is a y-term and a -term. But notice that there is no x-term, only an -term. We have seen this before and know that it means h is 0. We will need to complete the square for the y terms, but not for the x terms.

ⓐ Find the center and radius, then ⓑ graph the circle:

We need to rewrite this general form into standard form in order to find the center and radius.

Group the x-terms and y-terms.
There are no constants to collect on the right side.
Complete the square for
Rewrite as binomial squares.
Identify the center and radius.	Center: radius: 4
Graph the circle.

Key Concepts

• Distance Formula: The distance d between the two points and is
  

  

• Midpoint Formula: The midpoint of the line segment whose endpoints are the two points and is
  

  

  
  To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.

• Circle: A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the center, and the fixed distance is called the radius, r, of the circle.
• Standard Form of the Equation a Circle: The standard form of the equation of a circle with center, and radius, r, is
  
• General Form of the Equation of a Circle: The general form of the equation of a circle is
  

  

Glossary

circle

A circle is all points in a plane that are a fixed distance from a fixed point in the plane.