Question

In the coordinate plane, the point A, 4− 2 is translated to the point A′, 1− 5. Under the same translation, the points B, 7− 4 and C, 2− 5 are translated to B′ and C′, respectively. What are the coordinates of B′ and C′?

Answer 1

The coordinates of B' and C' are
`B'(x,y) = (4, -7)` and
`C'(x,y) = (-1, -8)`, respectively.

Explanation:

From the Linear Algebra, we define the translation of a given point as:

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

Where:

`O(x,y)` - Original point, dimensionless.

`T(x,y)` - Translation vector, dimensionless.

`O'(x,y)` - Translated point, dimensionless.

If we know that
`A'(x,y) = (1, -5)` and
`A(x,y) = (4,-2)`, then the translation vector is:

`T(x,y) = A'(x,y)-A(x,y)` (2)

`T(x,y) = (1,-5)-(4,-2)`

`T(x,y) = (-3,-3)`

If we know that
`B(x,y) = (7,-4)`,
`C(x,y) = (2,-5)` and
`T(x,y) = (-3,-3)`, then the translated points are, respectively:

`B'(x,y) = B(x,y)+T(x,y)` (3)

`B'(x,y) = (7,-4) +(-3,-3)`

`B'(x,y) = (4, -7)`

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

`C'(x,y) = (2,-5) + (-3,-3)`

`C'(x,y) = (-1, -8)`

The coordinates of B' and C' are
`B'(x,y) = (4, -7)` and
`C'(x,y) = (-1, -8)`, respectively.