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# Distance Formula

## Using the Pythagorean Theorem to determine distances

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SF Map - PythThm
Teacher Contributed

## Real World Applications – Algebra I

### Topic

How can we calculate the distances between specific places on a map?

### Student Exploration

Below is a screen shot of a part of San Francisco, California.

Let’s practice plotting points and writing ordered pairs based on this map.

1. First, we need our horizontal and vertical axes. Draw your x\begin{align*}x-\end{align*}axis on 24th\begin{align*}24^{th}\end{align*} Street, and draw your y\begin{align*}y-\end{align*}axis on Mission Street. Every unit on the x\begin{align*}x-\end{align*}axis will be one horizontal block, and every unit on the y\begin{align*}y-\end{align*}axis will be one vertical city block. Draw and label your tick marks on each axis.
2. What is at the origin? Label that point on your map.
3. What is the ordered pair for where “El Capitan” is located?
4. City College of San Francisco is located on Valencia St. between 22nd\begin{align*}22^{nd}\end{align*} and 23rd\begin{align*}23^{rd}\end{align*} Avenues. Mark a point on this map and label with the ordered pair and the name of the college.
5. John O’Connell High School is the brown colored block located between Folsom St, Harrison St, 19th\begin{align*}19^{th}\end{align*} St and 20th\begin{align*}20^{th}\end{align*} St. How would you describe this point as an ordered pair?
6. Make a right triangle using the streets between John O’Connell High school and City College. Now that we have a right triangle, use the Pythagorean Theorem to find the distance between the two schools.
7. Using the ordered pairs of the two schools, use the distance formula to find the distance between the two schools.
8. Are your answers for #6 and #7 the same? Why or why not?

We can also calculate distance between two major points on the map by using the midpoint formula. How do you think the midpoint formula can be useful when looking at maps?

Find two points on the map that we’d want to find the midpoint between. We can choose John O’Connell High School and City College. You can use the concept to help you determine how to find the midpoint between those two landmarks.

### Extension Investigation

9. Create a map of the city or town you live in. Determine where your x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}axes are and draw them on your map.
10. Draw and label some of the landmarks you feel are important on your map. Label the ordered pairs.
11. Draw and label the other schools on your map. Include all elementary schools, preschools, even colleges!
12. Draw and label all of the liquor stores on your map. You might want to mark these landmarks in a different color or with a different symbol.
13. What similarities can you make about the number of and locations of all of the schools in the area with the number of and locations of all of the liquor stores? Do you see any differences?
14. Find some of the distances between some of the schools using either the Pythagorean Theorem or the distance formula. Record your data on a table.
15. Find some of the distances between some of the schools using either the Pythagorean Theorem or the distance formula. Record your data on a table.
16. What similarities and differences do you notice between your two data tables? What conclusions can you draw about your observations?
17. Find the midpoint between some of the landmarks you have chosen. What can the midpoints tell you about your map? What can this tell you about the area you chose?

### Connections to other CK-12 Subject Areas

#### Geometry

• Pythagorean Theorem

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