Question

FG has a midpoint at M(6, 2). Point G is at (8, 3). Find the coordinates of point F.

Answer 1

Answer:

F = (4, 1)

Step-by-step explanation:

The coordinates of a Midpoint

Once we understand how the coordinates of the midpoint of a segment are found, it'll be easier for us to understand how to approach this problem.

Given some two points in a cartesian plane (points that can be defined using x-y value pairs), A = (x1, y1) and B = (x2, y2), the midpoint, which we'll call M, of segment AB is structured as follows:

$M = \left(\frac{x_1 + x_2}2, \frac{y_1 + y_2}2\right)$

In a way, we think of the midpoint as the average between the two endpoints. Thus, its x value is the average of the endpoints' x values, and so is its y value.

Finding the endpoint of a segment given an endpoint and the midpoint

Now knowing how a midpoint is structured, we better understand how to approach a problem where our unknown is the endpoint of a segment.

We can define this endpoint as some x-y value pair, F = (x1, y1).
We know that another endpoint for this segment is point G = (8, 3). If we compare this to the explanation about the coordinates of a midpoint, we can say that:

$G = (x_2, y_2) = (8,3)\\\to x_2 = 8\text{, }y_2=3$

Finally, we are also given the coordinates of the midpoint, M, where M = (6, 2). If we also compare this to the previous explanation, we can say that:

$M = \left(\frac{x_1 + x_2}2,\frac{y_1+y_2}2\right) = (6,2)\\\\\to \frac{x_1+x_2}2=6\\\\\to \frac{y_1+y_2}2=2$

If we substitute G's coordinates, x2 = 8 and y2 = 3, into the two equations, we get two neat equations for x1 and y1 (F's coordinates) respectively:

$\frac{x_1 + 8}2=6\\\\\frac{y_1+3}2=2$

We'll solve each of the equations for x1 and y1 respectively.

$\to \frac{x_1 + 8}2=6\text{ //}\times2\\x_1 + 8 = 12\text{ //}-8\\x_1 = 4\\\\\to \frac{y_1 + 3}2 = 2\text{ //}\times2\\y_1 + 3 = 4\text{ //}-3\\y_1 = 1$

F's coordinates are (4, 1).