Question

XYZ has coordinates X(2, 3), Y(1,4), and Z(8,9). A translation maps X to X'(4,7). What are the coordinates for Y' and Z' for this translation?​

Answer 1

The coordinates are
`Y'(x,y) = (3, 8)` and
`Z'(x,y) = (10, 13)`.

Explanation:

First, we have to derive an expression for translation under the assumption that each point of XYZ experiments the same translation. Vectorially speaking, translation from X to X' is defined by:

`X'(x,y) = X(x,y) + T(x,y)` (1)

Where
`T(x,y)` is the vector translation.

If we know that
`X(x,y) = (2,3)` and
`X'(x,y) = (4,7)`, then the vector translation is:

`T(x,y) = X'(x,y)-X(x,y)`

`T(x,y) = (4,7) - (2,3)`

`T(x,y) = (2, 4)`

Then, we determine the coordinates for Y' and Z':

`Y'(x,y) = Y(x,y) + T(x,y)`

`Y'(x,y) = (1,4) + (2,4)`

`Y'(x,y) = (3, 8)`

`Z'(x,y) = Z(x,y) + T(x,y)`

`Z'(x,y) =(8,9) + (2,4)`

`Z'(x,y) = (10, 13)`

The coordinates are
`Y'(x,y) = (3, 8)` and
`Z'(x,y) = (10, 13)`.