Hi! I'm in grade 7, I need a simpler method to know how to solve this:   A line segment ST has midpoint M=(-2,4) and the coordinates of point T are (1,9). Calculate the coordinates of point S.

Expert Answers
ishpiro eNotes educator| Certified Educator

The most straightforward method is to use the midpoint formula. The coordinates of the midpoint M, `(x_M, y_M)` can be found as

`x_M = (x_S + x_T)/2` , `y_M = (y_S + y_T)/2`

Now, we do know the coordinates of the point M, but we do not know the coordinates of point S. We can plug in everything we know into the formulas and then solve for the unknowns.

For x-coordinate: `-2 = (x_S + 1)/2`

Multiply both sides by 2: `-4 = x_S + 1`

Subtract 1 from both sides: `-5 = x_S` , or `x_S = -5` .

Similarly, for y-coordinate: `4 = (y_S+9)/2`

From here, `8 = y_S + 9` and `y_S = -1` .

So point S has coordinates (-5, -1).

An alternative method to figure this out is to consider that since the midpoint divides the segment in half, it is the same distance from point S and point T.

The horizontal distance between point M and point T is the difference of x-coordinates: there are 3 units between `x_M = -2` and `x_T = 1` . Then, the x-coordinate of S must be 3 units away from `x_M` : -2 - 3 = -5.

So `x_S = -5.`

In the y direction, point M is 5 units away from T: `y_M = 4` and `y_T = 9` .

So y-coordinate of S will be 5 less than `y_M` : 4 - 5 = -1. `y_S = -1` .

Again, point S has coordinates (-5, -1).

Depending on your personal preference, one method might seem simpler than another.