How Far Between Cities?
Part II

Student Page

Activity: How can you find the distance (in miles) between Rialto and Loma Linda using this map?

Note: If you need more review about how the triangles are important, you can review Part I again.

To look at the map, Open the Java Applet

Note: It will open in a separate window. Arrange your browser windows so that the applet and the student page can be easily viewed.

Here are some things to notice on the map:

  • display of x (in pixels)
  • display of y (in pixels)
  • scale of pixels to miles

NOTE: A pixel is the basic unit of the composition of an image on a computer monitor (usually a single-colored dot).

Assuming that care has been taken to have horizontal and vertical line segments, here are some questions to think about:

Looking at the x-coordinates
  1. Click on Rialto. What value is x?
  2. Click on Loma Linda. What value is x?
  3. What is the difference between the two values of x?
Looking at the y-coordinates
  1. Click on Rialto. What value is y?
  2. Click on Loma Linda. What value is y?
  3. What is the difference between the two values of y?
Suppose we drew some lines on the map:

  1. Does the horizontal line segment represent the difference between the x-coordinate of Rialto and the x-coordinate of Loma Linda? How long is it?
  2. Does the vertical line segment represent the difference between the y-coordinate of Rialto and the y-coordinate of Loma Linda? How long is it?
Pythagorean Theorem:
  1. What is the Pythagorean Theorem? [Check this Ask Dr. Math FAQ.]
  2. Use the Pythagorean Theorem to calculate the number of pixels between Rialto and Loma Linda.
Let's look at the hypotenuse of that triangle:

If you know how many pixels the distance (as the "crow" flies) between Rialto and Loma Linda, can you convert that to miles?

Converting pixels to miles:
  1. At the top of the Java applet page, this scale is given:

    25 pixels = 1 mile

  2. Use these ratios to convert pixels to miles:

    [Need to understand more about ratios? Check this Ask Dr. Math archive:Figuring Ratios.]

  3. If you use the number of pixels that you have already calculated in this ratio than you should be able to do some arithmetic to find the distance in miles.

Practice: Find the distances between these other cities in the Inland Empire including:

Fontana and San Bernardino
Muscoy and Loma Linda


Generalize the method that you used to find the distances between the three pairs of cities. Write the steps needed to calculate the distance between any two cities on the map.

Back to Part I


Selections from Ask Dr. Math Archives:

30-60-90 and 45-45-90 Triangles
If I have a triangle that is 30-60-90 or 45-45-90, how do I find all the sides when given only one side?

Defining Distance Mathematically
What is wrong with D' = sqrt(X^2 - X'2)?

FAQ: The Pythagorean Theorem
What is the Pythagorean theorem? When would you use it? How can we prove it?

Figuring Ratios
How do you figure the ratio of something?

Finding a Point Equidistant From Two Other Points
Point A is (-5,-3), and point B is (-1,-5); to be equidistant from A and B, what should the value of k be for the point (3,k)?

First Math Teacher
Who was the first math teacher? How did Pythagoreas come up with a^2 + b^2 = c^2?

How Long is the Hypotenuse?
In a right triangle, the lengths of the segments connecting the points of trisection of the hypotenuse to the vertex of the right angle are 7 and 9...

Midpoint of a Straight Line Segment
What is the midpoint of this: (-3,4) (5,-4)? Use the distance formula.

Ratio and Proportion: Beaches and Hawks
On a map, the scale states that 3 inches represent 125 miles. Two beaches are 5.2 inches apart. How far apart are they in miles?

What does the second number in a ratio stand for?

Teaching about Bearings
Can you help us learn how to locate a point using two bearings?

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