# 10.2: Distance and Midpoint

**At Grade**Created by: CK-12

## 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 \begin{align*}AB\end{align*}" is written \begin{align*}\overline{AB}\end{align*}. The *length* of a segment is named by giving the endpoints *without* using an **overbar.** For example, the length of \begin{align*}\overline{AB}\end{align*} is written \begin{align*}AB\end{align*}. In some books you may also see \begin{align*}m \overline{AB}\end{align*}, or *measure* of \begin{align*}\overline{AB}\end{align*}, which means the same as \begin{align*}AB\end{align*}, that is, it is the *length* of the segment with endpoints \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

**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:

\begin{align*}3 \ inches \cdot \frac{2 \ miles}{1 \ inch} = 6 \ miles\end{align*}

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:

\begin{align*}|3-8| = |-5| = 5\end{align*}

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: \begin{align*}|x|\end{align*}

## Distances on a Grid

In algebra you most likely worked with graphing lines in the \begin{align*}x-y\end{align*} **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 \begin{align*}x\end{align*}-**coordinates.** - If the two points line up
*vertically,*look at the change of value in the \begin{align*}y\end{align*}-**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 \begin{align*}x-\end{align*}**coordinate** of 2), so we can look at the *difference* in their \begin{align*}y-\end{align*}**coordinates:**

\begin{align*}|9-3| = |6| = 6\end{align*}

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 \begin{align*}y-\end{align*}**coordinate** of 4), so we can look at the *difference* in their \begin{align*}x-\end{align*}**coordinates.** Remember to take the **absolute value** of the *difference* between the values to find the distance:

\begin{align*}|-4-3| = |-7| = 7\end{align*}

The distance between the two points is 7 units.

**Reading Check:**

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

\begin{align*}\; \; \; \;\end{align*}

\begin{align*}\; \; \; \;\end{align*}

\begin{align*}\; \; \; \;\end{align*}

\begin{align*}\; \; \; \;\end{align*}

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

\begin{align*}\; \; \; \;\end{align*}

\begin{align*}\; \; \; \;\end{align*}

\begin{align*}\; \; \; \;\end{align*}

\begin{align*}\; \; \; \;\end{align*}

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

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

## The Distance Formula

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

If we put this triangle in a **coordinate plane** so \begin{align*}A\end{align*} has coordinates of \begin{align*}(x_1 , y_1)\end{align*} and \begin{align*}B\end{align*} has coordinates of (\begin{align*}x_2 , y_2\end{align*}), 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 \begin{align*}AC\end{align*} is \begin{align*}|x_2 - x_1|\end{align*} and the length of \begin{align*}BC\end{align*} is \begin{align*}|y_2- y_1 |\end{align*}

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,

\begin{align*}c^2 = |x_2- x_1 |^2 + |y_2- y_1 |^2\end{align*}

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,

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

We say that \begin{align*}c\end{align*} is the distance between the points \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, and we call the formula above the *Distance Formula.*

**Reading Check:**

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

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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.*

\begin{align*}{\; \; \;}\end{align*}

\begin{align*}{\; \; \;}\end{align*}

\begin{align*}{\; \; \;}\end{align*}

\begin{align*}{\; \; \;}\end{align*}

## 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 \begin{align*}B\end{align*} is the **midpoint** of \begin{align*}\overline{AC}\end{align*} since \begin{align*}\over{AB}\end{align*} is *congruent* to \begin{align*}\overline{BC}\end{align*}:

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:

\begin{align*}10 \div 2 = 5\end{align*}

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 \begin{align*}x-\end{align*}**coordinates** and the \begin{align*}y-\end{align*}**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 \begin{align*}(x_1 , y_1)\end{align*} and \begin{align*}(x_2 , y_2)\end{align*}:

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

Therefore, the **midpoint** is at:

**Reading Check:**

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

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

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

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

3. *What does the word equidistant mean?*

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

4. *How many midpoints can a line segment have?*

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

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

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

\begin{align*}\; \; \;\end{align*}

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