Geometry midpoint of a segment

Question

The midpoint of segment AB is (-2,5). The coordinates of point A are (-7,3). Find the coordinates of point B.

Answer 1

I can think of two ways to get this answer. The first is what I'd call the "textbook method." This is to use the midpoint formula and solve for the unknown x₁ and y₁ values. It would look like this:

(x₁ + (-7)) / 2 = -2 (y₁ + 3) / 2 = 5

x₁ + (-7) = -4 y₁ + 3 = 10

x₁ = 3 y₁ = 7

But when the midpoint and an endpoint are given, there is another intuitive method that is really quick. I'm going to place the points in the order that you'd find them if you were going from point A to point B through midpoint M.

A (-7, 3)

M (-2, 5)

B (__, __)

The x value from A to M increased by 5. So it will increase by 5 more between M and B. -2 + 5 = 3. The y value from A to M increased by 2. So it will increase by 2 more between M and B. 5 + 2 = 7. This gets us the same answer (3, 7). We could verify that the coordinated for A and B would get this midpoint if we use the formula (which is effectively just averaging each pair of x and y values).

Answer 2

Let B = (x, y)

Since (-2, 5) is the midpoint of the line segment with endpoints (x, y ) and (-7, 3), we have:

(x + (-7)) / 2 = -2 and (y + 3) / 2 = 5

So, x - 7 = -4 and y + 3 = 10

x = 3 and y = 7

B = (3, 7)