Distance on the coordinate plane
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Milena's town is built on a grid similar to the coordinate plane. She is riding her bicycle from her home at point negative 3, 4 to the mall at point negative 3, negative 7. Each unit on the graph denotes one city block. Plot the two points, and find the distance between Milena's home and the mall. So let's see, she's riding her bicycle from her home at the point negative 3, 4. So let's plot negative 3, 4. So I'll use this point right over here. So negative 3 is our x-coordinate. So we're going to go 3 to the left of the origin 1, 2, 3. That gets us a negative 3. And positive 4 is our y-coordinate. So we're going to go 4 above the origin. Or I should say, we're going to go 4 up. So we went negative 3, or we went 3 to the left. That's negative 3, positive 4. Or you could say we went positive 4, negative 3. This tells us what we do in the horizontal direction. This tells us what we do in the vertical direction. That's where her home is. Now let's figure out where the mall is. It's at the point negative 3, negative 7. So negative 3, we went negative 3 along the horizontal direction and then negative 7 along the vertical direction. So we get to negative 3, negative 7 right over there. And now we need to figure out the distance between her home and the mall. Now, we could actually count it out, or we could just compute it. If we wanted to count it out, it's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 blocks. So we could type that in. And another way to think about it is they have the exact same x-coordinate. They're both at the x-coordinate negative 3. The only difference between these two is what is happening in the y-coordinate. This is at a positive 4. This is at a negative 7. Positive 4, negative 7. So we're really trying to find the distance between 4 and negative 7. So if I were to say 4 minus negative 7, we would get this distance right over here. So we have 4 minus negative 7, which is the same thing as 4 plus 7, which is 11. Let's do a couple more. Carlos is hanging a poster in the area shown by the red rectangle. He is placing a nail in the center of the blue line. In the second graph, plot the point where he places the nail. So he wants to place a nail in the center of the blue line. The blue line is 6 units long. The center is right over here. That's 3 to the right, 3 to the left. So he wants to put the nail at the point x equals 0, y is equal to 4. So he wants to put it at x is equal to 0, y is equal to 4. That's this point right over here. So let's check our answer. Let's do one more. Town A and Town B are connected by a train that has a station at the point negative 1, 3. I see that. The train tracks are in blue. Fair enough. Which town is closer to the station along the train route, Town A or Town B? So they're not just asking us what's the kind of crow's flight, the distance that if you were to fly. They're saying, which town is closer to the station along the train route? So if you were to follow the train route just like that. So the A right over here, A if you were going along the train route, you would have to go 1, 2, 3, 4, 5, 6 in the x-direction, and then 1, 2, 3, 4, 5 along the y-direction. So you'd have to go a total of 11. If you're going from B, you're going 1, 2, 3, 4, 5, 6, 7, 8 9, 10, 11 along the x-direction, and then 1, 2, 3, 4, 5, 6, so 6 along the vertical direction. So you're going to go a total 17. So it's pretty obvious that A is closer along the tracks. Now, you could also think about it in terms of coordinates because A is at the coordinate of negative 7, 8. And if you were to think about negative 7, 8, to get from negative 7 to negative 1 along the x-coordinate, you're going to go 6. And then to go from 8 to 3, you're going to go 5 more. So you could also not necessarily count it out, you can actually just think about the coordinates. But either way, you see that town A is closer.