Question

Triangle XYZ has vertices X(8, −2.3), Y(6.5, 5), and Z(6, 3). When translated, X′ has coordinates (3.8, −0.3). Enter a rule to describe this transformation. Then find the coordinates of Y′ and Z′. Drag and drop each number into the correct box to complete the statements. The rule is (x, y) arrowright parenleftzex − , y + parenrightze. The coordinates are Y′parenleftze, parenrightze and Z′parenleftze, parenrightze.

Answer 1

The coordinates of
`Y'(x,y)` and
`Z'(x,y)` are
`Y'(x,y) = (2.3, 7)` and
`Z'(x,y) = (1.8, 5)`, respectively.

Explanation:

A translation is a geometrical operation consisting in moving a point a given distance. We proceed to describe the operation:

`X(x,y) +U(x,y) = X'(x,y)` (Eq. 1)

Where:

`X(x, y)` - Initial point on the cartesian plane, dimensionless.

`X'(x, y)` - Translated point on the cartesian plane, dimensionless.

`U(x, y)` - Translation component, dimensionless.

Vectorially speaking, we find that translation component is:

`U(x, y) = X'(x,y) -X(x,y)`

If we know that
`X(x,y) = (8, -2.3)` and
`X'(x,y) = (3.8, -0.3)`, the translation component is:

`U(x,y) = (3.8,-0.3)-(8,-2.3)`

`U(x,y) = (3.8-8, -0.3+2.3)`

`U(x,y) = (-4.2, 2)`

Now we determine the coordinates of
`Y'(x,y)` and
`Z'(x,y)`:

(
`Y(x,y) = (6.5, 5)`,
`Z(x,y) = (6,3)`)

`Y'(x,y) = Y(x,y) + U(x,y)` (Eq. 2)

`Y'(x,y)=(6.5, 5) + (-4.2, 2)`

`Y'(x,y) = (2.3, 7)`

`Z'(x,y) = Z(x,y) + U(x,y)` (Eq. 3)

`Z'(x,y) = (6,3)+(-4.2,2)`

`Z'(x,y) = (1.8, 5)`

The coordinates of
`Y'(x,y)` and
`Z'(x,y)` are
`Y'(x,y) = (2.3, 7)` and
`Z'(x,y) = (1.8, 5)`, respectively.