Question

When you are trying to find the endpoint when given the midpoint and one point, what would the new formula be? Manipulate the formula using skills for solving literal equations. How would you find LaTeX: x_2\:\:\:or\:\:\:y_2x 2 o r y 2? Use LaTeX: x_m\:\:and\:\:\:\:y_mx m a n d y mto notate the midpoints. Remember the original midpoint formula is LaTeX: \left(\frac{x_1+\:x_2}{2},\:\:\frac{y_{1\:}+y_2}{2}\right)\:=\:\left(x_m,\:y_m\right)

Answer 1

`(x_2,y_2) = (2x_m - x_1 ,2y_m - y_1 )`

Explanation:

Given

`\left((x_1+\:x_2)/(2),\:\:(y_(1\:)+y_2)/(2)\right)\:=\:\left(x_m,\:y_m\right)`

Required

Determine x2, y2

Start by splitting the expression

`x_m = \left((x_1+\:x_2)/(2))` and
`y_m = ((y_(1\:)+y_2)/(2))`

Solving for x2 in
`x_m = \left((x_1+\:x_2)/(2))`

Multiply through by 2

`2 * x_m = (x_1 + x_2)/(2) * 2`

`2x_m = x_1 + x_2`

Make x2 the subject;

`x_2 = 2x_m - x_1`

Similarly:

`y_m = ((y_(1\:)+y_2)/(2))`

Multiply through by 2

`2 * y_m = (y_1 + y_2)/(2) * 2`

`2y_m = y_1 + y_2`

Make y2 the subject;

`y_2 = 2y_m - y_1`

Hence:

`(x_2,y_2) = (2x_m - x_1 ,2y_m - y_1 )`