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What is the distance between the points 6, -7) and (1, 5)?

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2 Answers

Hi Lar,

Mykola is right that you can get the answer to this problem using the distance formula -- but simply plugging-and-chugging numbers in a formula doesn't teach you anything. Why does the distance formula work? Where in the world does that weird expression come from?

To see why the distance formula works, plot the two points (6,-7) and (1,5) on the coordinate plane. We'd like to find the distance between the two points -- but what does this mean? It means we want to find the length of the line segment that connects those two points. So on your picture, draw a line connecting the two points.

Now how can we figure out the length of this line? The trick is to realize that we could draw a right triangle having the line segment we just drew as its hypotenuse. (In fact, there are actually two right triangles we could draw: one above the hypotenuse, and one below it.) Why does drawing a right triangle help us find the distance between our two points? Because of the Pythagorean Theorem! Once we know the lengths of the legs of our right triangle, we can use the Pythagorean Theorem to find the length of the hypotenuse, which is what we want.

So to finish the problem, we just have to find the lengths of the legs. See if you can figure out what those lengths will be from your drawing. If you want to check your work, leave a comment and I'll respond.

Good luck!

Matt L

Comments

This problem can be done lot simpler. The absolute change in x-direction is 5, and in y-direction is 12. By Pythagorean triple 5-12-13, the distance is 13.

1/24/2013

Robert, your comment doesn't explain anything -- it doesn't teach the student how to find distances in general, or where the distance formula comes from. You're right, of course, that it happens to work out in this problem that the distance is 13. But simply telling the student the answer, or pointing out the happy coincidence that the side lengths of the triangle form a Pythagorean triple, doesn't teach him anything; it merely shows that you know how to do the problem.

1/24/2013
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This is a question that requires the distance formula:

d=√[(x2-x1)2+(y2-y1)2]

Make the coordinates of one point x1 and y1, and the other point's coordinates x2 and y2. Plug them in and solve. Here's the set up:

d=√[(6-1)2+(-7-5)2] or d=√[(1-6)2+(5-(-7))2]. It's important to keep the x1 and x2 as well as the y1 and y2 coordinates in order.

Comments

Just to add to this in the case of confusion, the coordinate points you gave were in the format (x1,y1) (x2,y2).

1/24/2013
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