Question

One endpoint of a line segment has coordinates represented by (x+6, 1/3y). The midpoint of the line segment is (2,−5). How are the coordinates of the other endpoint expressed in terms of x and y?

Answer 1

Answer:

The coordinates of the other endpoint are $B(x,y) = \left(-x-2, -10-\frac{1}{3}\cdot y\right)$.

Step-by-step explanation:

Let $AB$ a line segment in which $M$ is the midpoint. If both $A$ and $M$ are given, then we determine the location of $B$ from definition of midpoint. That is:

$\frac{1}{2}\cdot \vec A + \frac{1}{2}\cdot \vec B = \vec M$ (Eq. 1)

Where:

$\vec A$, $\vec B$ - Endpoints with respect to origin, dimensionless.

$\vec M$ - Midpoint with respect to origin, dimensionless.

$\vec A + \vec B = 2\cdot \vec M$

$\vec B = 2\cdot \vec M - \vec A$

If we know that $A(x, y) = \left(x+6,\frac{1}{3}\cdot y \right)$ and $M(x, y) = (2,-5)$, then the coordinates of $\vec B$ are:

$\vec B = 2\cdot (2,-5)-\left(x+6,\frac{1}{3}\cdot y \right)$

$\vec B = (4,-10)-\left(x+6,\frac{1}{3}\cdot y \right)$

$\vec B = \left(-x-2, -10-\frac{1}{3}\cdot y\right)$

The coordinates of the other endpoint are $B(x,y) = \left(-x-2, -10-\frac{1}{3}\cdot y\right)$.