Question

Find the coordinates of the midpoint of MN with endpoints M = (2, 3) and N = (2, 9)

Answer 1

`(2, 6)`

Explanation:

Midpoint Formula:

The midpoint can be thought of as drawing a line between two points and then going half-way along that line, wherever that point is, is the midpoint. The vertical and distance will be cut in half.

I attached a diagram which I will be referring to in this explanation.

The main takeaway, is we have a formula, or at least somewhat of a formula:

`\text{midpoint} = (x_1 + \frac{\text{horizontal distance}}{2}, y_1 + \frac{\text{vertical distance}}{2})`

But if you look at the graph, you'll notice, the vertical and horizontal distance are simply the difference in x and y values.

`\text{horizontal distance} = x_2 - x_1\\\text{vertical distance} = y_2 - y_1`

This means we can substitute something into the equation.

`\text{midpoint} = (x_1 + (x_2-x_1)/(2), y_1 + (y_2-y_1)/(2))`

Now we could directly use this equation, but we can also simplify a bit, which we can do by rewriting the following:
`x_1\to (2x_1)/(2), y_1 \to (2y_1)/(2)`

`\text{midpoint} = ((2x_1+x_2-x_1)/(2),(2y_1+y_2-y_1)/(2))`

combining like values, you get:

`\text{midpoint}=((x_1+x_2)/(2), (y_1+y_2)/(2)})`

And now we have a convenient formula we can use for this problem, which is the midpoint formula.

Applying the Midpoint Formula:

Now all that's left to do, is plug in x and y values, and since we're adding values, the order does not matter.

`\text{midpoint} = ((2+2)/(2),(3+9)/(2))\to (2, 6)`

so the answer is:
`(2, 6)`