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

(−2−x,−10−1/3y)

Explanation:

Answer 2

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

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:

`(1)/(2)\cdot \vec A + (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,(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,(1)/(3)\cdot y \right)`

`\vec B = (4,-10)-\left(x+6,(1)/(3)\cdot y \right)`

`\vec B = \left(-x-2, -10-(1)/(3)\cdot y\right)`

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