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The Nearest Mile Marker

I hope you've been cruising along through the course so far. Now, I'm going to throw you a little roadblock. Let's say your car breaks down at mile marker 210 while driving down Interstate I-80. I-80 runs from west to east in the state of Nevada, and the interstate cuts across the entire state. Since your car's broken down, you need to walk to the nearest town. Using the map in the car you determine that there's a gas station 21 miles south of mile marker 190. So, you know you're here and you're wondering what's the distance to mile marker 190?

The Nearest Mile Marker

The distance between the mile markers is 20 miles, great work if you found this number. We know this is correct, since we want to just count the number of miles between 190 and 210. So, we really just want to subtract this mile marker from this one. So, 210 minus 190 equals 20 miles. And you might have thought to subtract 210 from 190. That actually would have worked as well. You just want to make sure that you take the absolute value of that number. Since 190 minus 210 is negative 20, we want the positive result, or the absolute value of that number. So either way you think about it, you'll still have a distance of

Distance to the Gas Station

So, if you walk along the interstate, and then south to the town, how far would you walk? Write that answer here. Then I want you to think about, is this the shortest distance that you can walk? Yes or no.

Distance to the Gas Station

You would have to walk 41 miles in total. That's pretty far. First, you'd start at mile marker 210 where your car broke down. You'd walk 20 miles west, and then 21 miles south. So we would just add 20 miles and 21 miles, which is 41 miles. Hopefully you also figured out that this is not the shortest distance. The shortest distance between any two points is actually a straight line. Now we're going to assume that the region in this area is relatively at the same elevation. Meaning we're not going uphill and downhill a lot in any direction we go. This is why we can be fairly sure that this distance would be the shortest distance from mile marker 210 straight to the town.

The Shortest Route

Now even if you weren't sure if this was the shortest distance, I want you to find this actual distance. What is C? When you've figured it out, write that answer here.

The Shortest Route

The shortest distance here would be 29 miles. Good thinking if you found this number. We can actually use the Pythagorean theorem in this case, since we have a right triangle. We know that our town is due south of the interstate, and our interstate runs from west to east. So, this means that the interstate and this north south direction meet at a right angle. So, the distance between these two mile markers is one leg of our triangle. And this distance, 21 miles, is another leg of our right triangle. So, we can square this leg and add it to the square of this leg to get the square of our hypotenuse. So, we'll have a squared plus b squared equals c squared. When we square these numbers and find their sum, we'll get 841 equals c squared. And then taking the square root of both sides, we'll have 1c is equal to the square root of 841, which equals 29. Great solving if you applied the Pythagorean theorem in this situation.

Another Town

So, we know you're at mile marker 210. But before you put your map away, your friend sees another town located seven miles south of mile marker 234. So what's the distance to this second town? Keep in mind that you want to find the shortest distance to the town. When you think you have that answer, enter it here. Then, let me know if this town is closer than this town. If this town is closer, chose yes, otherwise choose no.

Another Town

We can find the distance from our broken down car to the town by setting up a right triangle. We know the distance between the mile markers is 24, since we can subtract the smaller mile marker from the larger one. So, this distance is distance from the interstate to the town. We're looking for this distance d. So, we'll have this leg squared plus this leg squared equals our hypotenuse squared. Squaring the legs, we'll get 576 plus 49 equals d squared. This sum is d. We only need the positive root of 625 since we are dealing with distance, so d is equal to 25. So, if you had to pick which town to walk to, it's actually show to walk to town two. You have to walk 4 miles less than you would if you walked all the way to town 1.

The Distance Between Two Points

So, notice that in both cases, we're finding the distance between two points, here and here, and here and here. When we found the distance between these two points, we set up a right triangle and found the horizontal distance between them, and the vertical distance between them. This allowed us to use the Pythagorean Theorem to find the hypotenuse or the diagonal between the two points. The same was true in this case for town 2. We found the distance horizontally between the mile markers. And then the distance vertically, or south to go to the town. We squared these two numbers and found their sum, and then took the square root to get the distance. We can do something similar if we wanted to find the distance between any two points. So, let's say we had these points, negative 7 and 3, and 5 and 8. We can think of drawing two interstates our roads, horizontally and vertically. And the distance between the two points is the diagonal, or the hypotenuse of the right triangle we just created. To find the vertical distance between the two points, we want to look at the y values. We can think about this vertical distance as starting at y equals 3. Then it goes all the way up to y equals 8. So, if we find the difference, or the amount between the y values, we'll get the vertical distance, which is 5 here. So, knowing this, what do you think would be the horizontal distance? What's this distance?

The Distance Between Two Points

Well, this distance would be 12 units, great thinking if you found it. Since to find this vertical distance, we figured out how far we climbed on the y axis. We went from y equals 3, up 5 units to y equals 8. That's how we knew that this distance was 5. So to find the horizontal distance. We're going to start at x equals negative 7, this x value, and run all the way to the right, to x equals positive 5, this x value. So, the horizontal distance between these two points would be from negative 7 to positive 5. That would be a total distance of 12 units. You can find this distance by counting the spaces from negative 7 to positive 5. Or you can subtract the 2 numbers, and make sure to take the absolute value, so you get a positive number in the end. Here I just did 5 minus negative 7, which equals positive 12. If you would've subtracted 5 from negative 7, you would've gotten negative 12. So you would've had to take the absolute value. So that way you get a positive distance, or just 12.

The Pythagorean Theorem Again

When we think about the distance between two points, we want to picture a right triangle. The hypotenuse is the distance between them, and then we have the horizontal distance as one leg and the vertical distance between the points as another leg. We can use this right triangle to solve for any distance D. So, what do you think is the distance between these two points? Write that answer here.

The Pythagorean Theorem Again

It turns out that the distance is 13 units. Great solving if you found it. We can use the Pythagorean theorem to set up an equation. We'll have this leg squared plus this leg squared equals our hypotenuse squared. Solving for d, we'll get d is equal to 13 units.

The Legs of The Triangle

But let's be more general. We can do this for any 2 points in the Cornet plane. So if this point is x1 y1 and this point is x2 y2, lets see if we can create a formula for the distance between them. Before we get to write an equation, I want you to write an two expressions. What's an expression for this horizontal distance? Write that here. And what's an expression for the vertical distance? Write that one here.

The Legs of the Triangle

We know that this distance here is this distance on the number line. So if we go all the way to x2 and subtract off x1, we'll be left with the part in between them. For the vertical distance this part we want to look at the y values. We start at y1 and then we climb all the way up to y2. So if we want to find this vertical distance, we really want to find this vertical distance on the number line. So, we'll go all the way up to y2, and we'll subtract off y1. This will leave us with just this distance here. So, this is how we know that the horizontal distance is x2 minus x1, and the vertical distance is y2 minus y1.

The Distance Formula

So now that we have the set up for a right triangle, I want you to find the equation for the distance. You can either write an equation set to d squared, or you can have an equation set equal to d. Good luck on this one.

The Distance Formula

One possible answer is this equation. We're going to square this leg of the triangle. Add it to the square of this leg of the triangle. Which is equal to d squared, hypotenuse. This is really just the Pythagorean theorem. We have a number squared. Plus another number squared. Equals they hypotenuse squared. And what we could do is take the square root of both sides of the equation. This would leave us with d equal to the square root of x2 minus x1 squared, plus y2 minus y1 squared. Great thinking if you got either of the equations. Now, the order of these variables doesn't actually matter, since we're squaring the numbers. If we wound up with a negative value here, it would become positive, since we're squaring it. The same is true for the y value. If this value is negative, and we squared it, it would wind up being positive. So, perhaps an easier way to think about it, is just, think about the change in the x and the change in the y. How far did we travel this way, and how far did we travel this way. We'll square both of those, take their sum, and then take the square root to find the distance. This symbol right here that you see here is delta and it’s a Greek letter that means change. You may have seen this in chemistry or in other courses. And this delta just means change. I think it’s easy to think about the change in the x values. And the change in the y values to find our answer. This is more simple than writing this crazy notation. Memorizing the formula is not so important as really understanding that we're setting up a right triangle in space, when we find the distance between two points. That can always help you problem solve.

Distance Practice 1

For your first practice problem try finding the distance between these two points. This point is at 3 comma 2 and this point is at negative 1 comma 8. When you think you have the distance, enter it here.

Distance Practice 1

The answer is 2 times the square root of 13. Great thinking if you found this number. We want to find the distance between these two points. So, we need to set up a right triangle with the vertical distance and the horizontal distance. The horizontal distance between x equals negative 1 and x equals positive 3 is at y equals 2, which is here, and then we go all the way up to y equals 8, which is here. So, this vertical distance on the y axis is a distance of 6 units. Once we've found these two legs of our right triangle, we can use the Pythagorean theorem to find the hypotenuse. So, we'll have this leg squared, plus this leg squared, equals d squared, our hypotenuse. We solve this equation to get d squared equals 52. And then we take the square root of both sides to get d is equal to the square root of 52. Simplifying the square root of 52, we'll get d is equal to 2 times root 13. This is our simplified distance. And again, notice that the horizontal distance is simply the change in the x values. We went from x equals negative 1 to x equals 3. And notice that the vertical distance was the change in the y values. We went from y equals 2 all the way up to y equals 8. So, we'll have the change in y squared, plus the change in x squared equals d squared. We can use our algebra skills and eventually solve for d.

Distance Practice 2

Try finding the distance between the points 2, negative 4 and negative 3, 6. Be careful when you find the change in the x values and the change in the y values. When you think have the distance, type it in here.

Distance Practice 2

The distance between the points is 5 times the square root of 5. Great solving if you found that one. This is a rough sketch of the points on the coordinate plane. Notice that I'm not precise, and that's okay. We're really just after the difference in the x values, and the difference in the y values, or the change in them. So, to find the distance between these two points, we need to set up our right triangle. First we'll find the horizontal distance. We'll go from x equals 2 to x equals negative 3. That's a distance of five units. Now if that's confusing, try thinking about it by starting at x equals negative 3. If we start at x equals negative 3 and count up to x equals 2, we will travel the five units horizontally. We want to think about the horizontal distance when traveling on a number line. Now we need to find the vertical distance. We start at y equals negative 4 which is here. And then we climb all the way up to y equals 6 here. The vertical distance between y equals negative 4 and y equals 6 is ten units. So we know this vertical distance is simply ten. Now that we have the legs of our right triangle, we can solve for the distance. We'll have the change in x squared which is 5 squared. the change in y squared, which is 10 squared equal to d squared, our hypotenus. We'll add the sum of the squares, to get d squared equals 125, and then, we'll square root both sides, to get 1d is equal to the square root of 25 times 5. We simplify this square root to get our final answer of 5 root 5. Again great work if you got some of that correct. Even more amazing if you got it right on your first try.

Distance Practice 3

For the last problem, find the distance between the point 3, 4 and negative 2, negative 3. Good luck.

Distance Practice 3

The distance between the two points is the square root of 74. Great problem solving if you found it. We'll start by drawing a rough sketch on the points on the coordinate plane. If we want to find this distance between the two points, we set up a right triangle in two dimensional space. The horizontal distance goes from x equals negative 2 to x equals positive 3. So, we know that this distance is a horizontal distance between the values of negative 2 and positive want to find the difference between the y values or the change in them. We start at y equals negative 3, and we climb all the way up to y equals 4. This distance, so we know that this distance is 7 units. Keep in mind that the distance on this vertical axis, or between the y coordinates, is the same out here. This is a vertical distance that's the same as this one. So, we'll have the change in x squared, which is 5 squared, plus the change in y squared, which is 7 squared equals d squared, our hypotenuse. Squaring these numbers, and finding this sum, will get 74 equals d squared. Now, we just take the square root of both sides, keeping only the positive root. So, one d is equal to the square root of 74. Now, if we try to simplify the square root of 74, we notice that its factors are 2 and 37. These are not perfect squares, so we just leave our answer as the square root of 74.