why can you consider the midpoint of a line segment on the number line to be the average of its coordinates?

Answer 1

Say you have the following points:

a < m < b

And you want the midpoint to be m. Then you know:

m - a = b - m

Since m is the same distance from a as it is from b.

Rearrange that equation:

m - a = b - m
m = a + b - m
2m = a + b

m = (a+b)/2

That's the average. So that's why. This works for any number of dimensions (you average each coordinate independently).

Answer 2

Any line segment can be defined by rectangular coordinates, meaning you can draw a triangle with the x dimension and y dimension as its legs.

The average of the x dimension will be its midpoint.
The average of the y dimension will be its midpoint.

You can now use these 2 references as coordinates to draw a smaller triangle that will also define the midpoint of your line segment.

Answer 3

You can translate your general line segment so one end is at (0,0), the other at (a,b). Midpoint, (x,y) is by definition EQUAL distance from each end. So
√[(x-0)² + (y-0)²] = √[(a-x)² + (b-y)²]
(x-0)² + (y-0)² = (a-x)² + (b-y)²
x² + y² = a² - 2ax + x² + b² - 2by + y²
0 = a² - 2ax + b² - 2by
2ax + 2by = a² + b²

Now there's an infinite number of points equidistant from each end of the segment -- all the points on the perpendicular bisector. We need our point to be ON the line through (0,0) and (a,b), which means y = (b/a)x. Plugging that in,
2ax + 2by = a² + b²
2ax + 2b(b/a)x = a²+b²
2ax + (2b²/a)x = a² + b²
(2a + 2b²/a)x = a² + b²
(2a² + 2b²)x/a = a²+b²
x/a = 1/2
x = a/2

y = (b/a)x = (b/a)(a/x) = b/2.

If you're really ambitious, try it without the translation to (0,0). But this should give you the idea.

Answer 4

Because the number exactly in the middle between two numbers is the average. The x-coordinate of the midpoint should be exactly in the middle between the two x-coodinates, and same for the y-coodinate of the midpoint.

Answer 5

because when you add all the numbers and divide by 2, that's your midpoint

Answer 6

b/c average means its middle value

Answer 7

This is true only for STRAIGHT lines!!!