Why does finding the average of two points give you the midpoint?

Cambridge101 said:
I am trying to see the link here. Can someone please explain!!
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If you have two tests in a course you are doing, say you saore [imath]84~\&~76[/imath] out of [imath]100[/imath] what is your average on the two?
Well is it [imath]\dfrac{84+76}{2}~?[/imath] The mid-point is the average of the end-points.

[imath][/imath][imath][/imath]
 
pka said:
If you have two tests in a course you are doing, say you saore [imath]84~\&~76[/imath] out of [imath]100[/imath] what is your average on the two?
Well is it [imath]\dfrac{84+76}{2}~?[/imath] The mid-point is the average of the end-points.

[imath][/imath][imath][/imath]
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This does not explain at all what I asked - I asked why!
 
Cambridge101 said:
This does not explain at all what I asked - I asked why!
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Suppose that [imath]a<b\\\dfrac{a}{2}<\dfrac{b}{2}\\\dfrac{a}{2}+\dfrac{a}{2}<\dfrac{b}{2}+\dfrac{a}{2}\\a<\dfrac{a+b}{2}[/imath] likewise [imath]\dfrac{a+b}{2}<b[/imath] which means [imath]a<\dfrac{a+b}{2}<b[/imath]
The average of two numbers is between them. Now prove it is equal distance.
 
Cambridge101 said:
I am trying to see the link here. Can someone please explain!!
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I think lev888 is right: In order to help you see the connection, we need to start with your understanding of what each concept means. For example, does "average", to you, just mean the formula (half the sum), or something more?

In particular, "why" is a tricky word; such a question can be answered in several very different ways. So we may need to make several attempts to explain, and your feedback will clarify what you think is lacking.

What does "why" or "the link" mean to you?
 
Dr.Peterson said:
I think lev888 is right: In order to help you see the connection, we need to start with your understanding of what each concept means. For example, does "average", to you, just mean the formula (half the sum), or something more?

In particular, "why" is a tricky word; such a question can be answered in several very different ways. So we may need to make several attempts to explain, and your feedback will clarify what you think is lacking.

What does "why" or "the link" mean to you?
Click to expand...
Well,

I would say the midpoint is a point on a line segment which is equidistant. Therefore say were going from 0 to 3 on the number line, the point 1.5 is the equidistant point (the midpoint), where to the distance needed to travel forward from 0 to reach 1.5 is the same distance needed to travel back from 3 to also reach 1.5, thus it has to central. In this case, 1.5 is the distance needed to travel from 0 to get to 1.5 and 1.5 is the distance needed to travel backwards from 3 to get to 1.5.

Thus, because 1.5 is the equidistant point, it is the only point on the line segment which both 0 and 3 can translate to by adding or taking away a common amount. What I mean by this is that i can add a common amount of 1.5 to 0 and take away this same common amount from 3 and 0 will become 1.5 while 3 also will now become 1.5. They both have traveled an equal distance of 1.5 (one forward and on back) and both reached the same point on the line.

The link I have attempted to draw from the average is this. The average takes these quantities 0 and 3 and shares the total amount being 0+3 = 3 equally (redistributes) this 3 into two equal quantities. The only possible quantity that can be possible to have in two equal groups to make 3 is 1.5. Almost like the 3 has lost 1.5 and given 0 a 1.5 in order for both quantities in these separate groups to be equal. This value happens to be 1.5.

Now I think I am getting close but I still cant make a solid connect. So does the reason finding the average of the total combined distance 0 and 3 and sharing it between 2 give us the midpoint. Because by sharing this quantity of 3 equally between 2, we are finding the only quantity that can exist which makes both quantities equal. In other words this equidistant point. Because it follows to reason that whatever 3 looses 0 must gain when finding this average. So when finding the average of the two points, we are actually finding the point which makes both groups equal when one has gained the same amount as the other has lost.

This is the only way I can put it into words rn…. I’m not very well.
 
I think you're pretty close to an understanding. I'll try saying it a couple other ways. (One thing I needed to see was whether you were specifically thinking of the formula for the midpoint of a line segment on a plane or in space (two or three dimensions), or were mostly thinking on a number line, as you say here. That will make it a little easier.

First, I'll talk about it symbolically. Rather than make one of the numbers by 0, let's take another example, say 3 and 15. The midpoint will be equidistant from these, so if it is x, then x-3 has to equal 15-x. We can solve that equation:

x - 3 = 15 - x​
x + x - 3 = 15 - x + x​
2x - 3 = 15​
2x - 3 + 3 = 15 + 3​
2x = 18​
2x/2 = 18/2​
x = 9​

As you look at that, you can see that we added 3 and 15, and divided by 2, which is how you average.

We can do the same thing with unknown numbers, a and b instead of 3 and 15:

x - a = b - x​
x + x - a = b - x + x​
2x - a = b​
2x - a + a = b + a​
2x = a + b​
2x/2 = (a + b)/2​
x = (a + b)/2​

Or we can do the same sort of thing, in reverse, starting with the average. As you say, the average is the number you get when you share the total equally (which is why you add and divide by 2). Is this number equidistant from the two given numbers? To find out, we can evaluate the differences:

(a + b)/2 - a = a/2 + b/2 - a = (b/2 + (a/2 - a) = b/2 - a/2 = (b - a)/2​
b - (a + b)/2 = b - a/2 - b/2 = (b - b/2) - a/2 = b/2 - a/2 = (b - a)/2​

So, the two distances are equal. And this reveals something useful: those distances are both half the difference between the given numbers.

And this, I think, is essentially what you came up with. An alternative way to find the average is to "split the difference". The average adds half the difference to the lower number, and subtracts half the difference from the higher number, so that it is the same distance from each.

Does that help at all?
 
Why in the world are you making complicated something so simple?

Think about the number line and point a and point b.

First, do you grasp that the distance between those points is |b - a|?

Now we have three possible cases.

a < b. In which case, the midpoint is

[math]a + \dfrac{|b - a|}{2} = a + \dfrac{b - a}{2} = \dfrac{2a}{2} + \dfrac{b - a}{2} = \dfrac{a + b}{2}.[/math]
a = b. In which case, the midpoint is

[math]a + \dfrac{|b - a|}{2} = a + \dfrac{0}{2} = a = \dfrac{2a}{a} = \dfrac{a + b}{2} \text { because } a = b.[/math]
a > b. In which case, the midpoint is

[math]a - \dfrac{|b- a|}{2} = a - \dfrac{a - b}{2} = a + \dfrac{b - a}{2} = \dfrac{2a}{2} + \dfrac{b - a}{2} = \dfrac{a + b}{2}.[/math]
 
I meant if a = b,

[math]a + \dfrac{|b - a|}{2} = a + \dfrac{0}{2} = a = \dfrac{2a}{2} = \dfrac{a + b}{2}.[/math]
 
Dr.Peterson said:
I think you're pretty close to an understanding. I'll try saying it a couple other ways. (One thing I needed to see was whether you were specifically thinking of the formula for the midpoint of a line segment on a plane or in space (two or three dimensions), or were mostly thinking on a number line, as you say here. That will make it a little easier.

First, I'll talk about it symbolically. Rather than make one of the numbers by 0, let's take another example, say 3 and 15. The midpoint will be equidistant from these, so if it is x, then x-3 has to equal 15-x. We can solve that equation:

x - 3 = 15 - x​
x + x - 3 = 15 - x + x​
2x - 3 = 15​
2x - 3 + 3 = 15 + 3​
2x = 18​
2x/2 = 18/2​
x = 9​

As you look at that, you can see that we added 3 and 15, and divided by 2, which is how you average.

We can do the same thing with unknown numbers, a and b instead of 3 and 15:

x - a = b - x​
x + x - a = b - x + x​
2x - a = b​
2x - a + a = b + a​
2x = a + b​
2x/2 = (a + b)/2​
x = (a + b)/2​

Or we can do the same sort of thing, in reverse, starting with the average. As you say, the average is the number you get when you share the total equally (which is why you add and divide by 2). Is this number equidistant from the two given numbers? To find out, we can evaluate the differences:

(a + b)/2 - a = a/2 + b/2 - a = (b/2 + (a/2 - a) = b/2 - a/2 = (b - a)/2​
b - (a + b)/2 = b - a/2 - b/2 = (b - b/2) - a/2 = b/2 - a/2 = (b - a)/2​

So, the two distances are equal. And this reveals something useful: those distances are both half the difference between the given numbers.

And this, I think, is essentially what you came up with. An alternative way to find the average is to "split the difference". The average adds half the difference to the lower number, and subtracts half the difference from the higher number, so that it is the same distance from each.

Does that help at all?
Click to expand...
Yes this was a great way of looking at it - very, very much appreciated.

Thank you!
 
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