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Learning Objectives

  • Derive the Distance Formula using the Pythagorean Theorem.
  • Use the Distance Formula to find the length of a line segment with known endpoints.
  • Use the Midpoint Formula to calculate the coordinates of the midpoint of a line segment given both endpoints, or to determine the coordinates of one endpoint given the midpoint and the other endpoint.

Measuring Distances

There are many different ways to identify measurements. This lesson will present some that may be familiar, and probably a few that are new to you.

Before we begin to examine distances, however, it is important to identify the meaning of distance in the context of geometry. The distance between two points is defined by the length of the line segment that connects them.

  • The distance between two points is the _________________________ of the line segment that connects them.

The most common way to measure distance is with a ruler. Also, distance can be estimated using scale on a map.

Notation Notes: When we name a segment we use the endpoints and an overbar (a bar or line above the letters) with no arrows. For example, "segment AB" is written \overline{AB}. The length of a segment is named by giving the endpoints without using an overbar. For example, the length of \overline{AB} is written AB. In some books you may also see m \overline{AB}, or measure of \overline{AB}, which means the same as AB, that is, it is the length of the segment with endpoints A and B.

Example 1

Use the scale to estimate the distance between Aaron’s house and Bijal’s house. Assume that the first third of the scale in black represents one inch.

You need to find the distance between the two houses in the map. The scale shows a sample distance. Use the scale to estimate the distance. You will find that approximately 3 segments of the length of the scale fit between the two points. Be careful — 3 is not the answer to this problem! As the scale shows 1 inch equal to 2 miles, you must multiply 3 units by 2 miles:

3 \ inches  \cdot  \frac{2 \ miles}{1 \ inch} = 6 \ miles

The distance between the houses is about six miles.

You can also use estimation to identify measurements in other geometric figures. Remember to include words like approximately, about, or estimation whenever you are finding an estimated answer.

The word “estimation” means using a non-exact guess of what a number is. Another similar word is “approximation.”

Both of these words are nouns. The verb forms are: “to estimate” or “to approximate.”

We use these words when we are not sure of the exact measurement of a distance, length, or other number, but when we can make an educated guess.

  • To estimate (or to _________________________________ ) a number means to give a non-exact but educated guess of what it is.

Rulers

You have probably been using rulers to measure distances for a long time and you know that a ruler is a tool with measurement markings.

  • A ruler is a tool with ___________________________________ markings.

Using a ruler: If you use a ruler to find the distance between two points, the distance will be the absolute value of the difference between the numbers shown on the ruler.

This means that you do not need to start measuring at the zero mark, as long as you use subtraction to find the distance.

Note: We say absolute value here since distances in geometry must always be positive, and subtraction can give a negative result.

  • You do not need to measure from zero on a ruler; just ______________________ the start number from the end number to find the distance!
  • The distance on a ruler is the ___________________________ value of the difference between the numbers.
  • Absolute value is always a ____________________________ number.

Example 2

What distance is marked on the ruler in the diagram below? Assume that the scale is marked in centimeters.

The way to use the ruler is to find the absolute value of the difference between the numbers shown. This means you subtract the numbers and then make sure your answer is positive. The line segment spans from 3 cm to 8 cm:

|3-8|   =   |-5|   =   5

The absolute value of the difference between the two numbers shown on the ruler above is 5 cm. So the line segment is 5 cm long.

Remember, we use vertical bars around an expression to show absolute value: |x|

Distances on a Grid

In algebra you most likely worked with graphing lines in the x-y coordinate plane. Sometimes you can find the distance between points on a coordinate plane using the values of the coordinates:

  • If the two points line up horizontally, look at the change of value in the x-coordinates.
  • If the two points line up vertically, look at the change of value in the y-coordinates.

The change in value will show the distance between the points. Remember to use absolute value, just like you did with the ruler. Later you will learn how to calculate distance between points that do not line up horizontally or vertically.

  • When points line up horizontally, they have the same _______-coordinate. This means their _______-coordinates are different so we take their difference to find the distance between the points.
  • When points line up vertically, they have the same _______-coordinate. This means their _______-coordinates are different so we take their difference to find the distance between the points.

Example 3

What is the distance between the two points shown below?

The two points shown on the grid are at (2, 9) and (2, 3). These points line up vertically (meaning they have the same x-coordinate of 2), so we can look at the difference in their y-coordinates:

|9-3|   =   |6|   =   6

So, the distance between the two points is 6 units.

Example 4

What is the distance between the two points shown below?

The two points shown on the grid are at (–4, 4) and (3, 4). These points line up horizontally (meaning they have the same y-coordinate of 4), so we can look at the difference in their x-coordinates. Remember to take the absolute value of the difference between the values to find the distance:

|-4-3|   =   |-7|   =   7

The distance between the two points is 7 units.

Reading Check:

1. What is absolute value? Explain in your own words.

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2. When 2 points line up vertically, what value do they have in common?

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3. When 2 points line up horizontally, what value do they have in common?

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The Distance Formula

We have learned that a right triangle with sides of lengths a and b and hypotenuse of length c has a special relationship called the Pythagorean Theorem. The sum of the squares of a and b is equal to the square of c. Placing this in equation form we have:

If we put this triangle in a coordinate plane so A has coordinates of (x_1 , y_1) and B has coordinates of (x_2 , y_2), we can find the lengths of the legs of the triangle using what we just learned about points that line up horizontally or vertically:

the length of AC is |x_2 - x_1| and the length of BC is |y_2- y_1 |

We are finding the length, which means that we want a positive value; the absolute value bars guarantee that our answer is always positive. But in the final equation,

c^2   =   |x_2- x_1 |^2    +   |y_2- y_1 |^2

the absolute value bars are not needed since we squared all three terms, and squared numbers are always positive.

Getting the square root of both sides we have,

c = \sqrt{(x_2- x_1 )^2 + (y_2- y_1 )^2}

We say that c is the distance between the points A and B, and we call the formula above the Distance Formula.

Reading Check:

1. On which famous theorem is the Distance Formula based?

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2. How can you find the distance of one of the legs of a right triangle like the one in the diagram on the previous page? Pick one leg and explain in your own words.

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Segment Midpoints

Now that you understand congruent segments, there are a number of new terms and types of figures you can explore.

A segment midpoint is a point on a line segment that divides the segment into two congruent segments. So, each segment between the midpoint and an endpoint will have the same length.

  • A midpoint divides a segment into two ___________________________ parts.

In the diagram below, point B is the midpoint of \overline{AC} since \over{AB} is congruent to \overline{BC}:

There is even a special postulate dedicated to midpoints:

Segment Midpoint Postulate

Any line segment will have exactly one midpoint—no more, and no less.

Example 5

Nandi and Arshad measure and find that their houses are 10 miles apart. If they agree to meet at the midpoint between their two houses, how far will each of them travel?

The easiest way to find the distance to the midpoint of the imagined segment connecting their houses is to divide the length (which is 10 miles) by 2:

10 \div 2  =  5

Each person will travel five miles to meet at the midpoint between their houses.

The Midpoint Formula

The midpoint is the middle point of a line segment. It is equidistant (equal distances) from both endpoints.

The formula for determining the midpoint of a segment in a coordinate plane is the average of the x-coordinates and the y-coordinates. Remember, to find the average of 2 numbers, you take the sum of the numbers and then divide by 2.

If a segment has endpoints (x_1 , y_1) and (x_2 , y_2):

  • the average of the x-coordinates is: \frac{x_1 + x_2}{2}
  • and the average of the y-coordinates is: \frac{y_1 + y_2}{2}

Therefore, the midpoint is at:

Reading Check:

1. What is an average? Explain in your own words.

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2. Where is a midpoint located on a line segment? Describe.

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3. What does the word equidistant mean?

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4. How many midpoints can a line segment have?

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5. In the space below, draw a line segment. Then draw and label its midpoint.

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