what are the midpoint coordinates for (6,7) and (4,3)

Question

it is geometry

 

Answer 1

Hi, Olivia:

 

1) Consider two points on the number line. As an example, let's take points x=0 and x=2. It is obvious, that the midpoint coordinate is x=1;

2) What if we have x=4 and x=9, for example? Again, the midpoint is x=6.5, since its distance from both points is the same. Indeed, |4-6.5|=|-2.5|=2.5 and |9-6.5|=|2.5|=2.5. Remember, distance is the absolute value of the difference between x-coordinates of the first and the second point.

3) Can we find a formula to obtain the x-coordinate of the midpoint? Let us try.

 

Let the first point be x₁ and the second be x₂. Let the midpoint be x_m. Without losing any generality, let us assume x₁<x₂. Then x₁<x_m<x₂, since midpoint shall be between two points. We know that distance from midpoint to each of two points is the same. This means that:

x_m-x₁=x₂-x_m; Solve for x_m,

 

2x_m=x₂+x₁; 

 

x_m=(x₁+x₂)/2;

 

So the coordinate of a midpoint is just the average of two coordinates.

 

Now, we can generalize. If we consider two points on the plane, with coordinates (x₁,y₁) and (x₂,y₂), it is natural to suggest that the coordinates of a midpoint will be given by the average of respective y- and x-coordinates of endpoints. So, we suggest that:

 

(x_m,y_m)=(½(x₁+x₂);½(y₁+y₂)); 

 

And this is indeed the case. So in your example,

 

x_m=½(6+4)=5;

y_m=½(7+3)=5;

 

or

 

(x_m,y_m)=(5,5)

 

As a matter of fact, this can be generalized for 3-dimensional case and for any number of dimensions.

Answer 2

mid point formula GO!

(x₂₊x₁)/2  gives the x coordinate

 

(y₁+y₂)/2 gives the y coordinate

 

(6,7) and (4,3)

 

lets make (6,7) point 1 

x₁ is 6

y1 is 7

 

 

lets make (4,3) point 2

 

x_{2 is} 4

y₂ is 3

 

when I was younger labeling the points like that helped me plug everything in correctly

 

so we have 

(4+6)/2  = 10/2 or 5

(7+3)/2 =10/2 or 5

 

so your midpoint is (5,5)