Question

ONM has vertices O (-4, 2), N (3, 6) and M(0,3). What are the coordinates of the

vertices of O' N' M' if ONM is first translated (x, y)+(x + 3, y) followed by a

reflection across the line y = 1?

Answer 1

The new vertices are
`O''(x,y) = (-1, 0)`,
`N''(x,y) = (6,-4)` and
`M''(x,y) = (3,-1)`.

Explanation:

Previously, we have to define a vectorial equation for a reflection across the line
`y = 1`:

Reflection across the line y = 1:

`P'(x,y) = P(x,y) -2\cdot [P(x,y) -(x_(P),1)]` (1)

Where:

`P(x,y)` - Original point.

`x_(P)` - x-Coordinate of the original point.

`P'(x,y)` - Resulting point.

Translation:

`P'(x,y) = P(x,y) + (3,0)` (2)

If we know that
`O(x,y) = (-4,2)`,
`N(x,y) = (3, 6)` and
`M(x,y) = (0,3)`, then the resulting points are, respectively:

Point O'

Translation

`O'(x,y) = O(x,y) + (3,0)`

`O'(x,y) = (-1,2)`

Reflection

`O''(x,y) = O'(x,y) -2\cdot [O'(x,y) -(x_(O'),1)]`

`O''(x,y) = (-1,2) -2\cdot [(-1, 2)-(-1,1)]`

`O''(x,y) = (-1, 0)`

Point N'

Translation

`N'(x,y) = N(x,y) + (3,0)`

`N'(x,y) = (6,6)`

Reflection

`N''(x,y) = N'(x,y) -2\cdot [N'(x,y) -(x_(N'),1)]`

`N''(x,y) = (6,6) - 2\cdot [(6,6)-(6,1)]`

`N''(x,y) = (6,-4)`

Point M'

Translation

`M'(x,y) = M(x,y) + (3,0)`

`M(x,y) = (3,3)`

Reflection

`M''(x,y) = M'(x,y) -2\cdot [M'(x,y) -(x_(M'),1)]`

`M''(x,y) = (3,3) - 2\cdot [(3,3)-(3,1)]`

`M''(x,y) = (3,-1)`

The new vertices are
`O''(x,y) = (-1, 0)`,
`N''(x,y) = (6,-4)` and
`M''(x,y) = (3,-1)`.