Question

What is the composition of the translations
(T〈−3, 4〉 ∘ T〈8, −7〉)(x, y) as one translation?

Answer 1

Given:

The composition of the translations is

`(T\langle-3,4\rangle \circ T\langle 8,-7\rangle )(x,y)`

To find:

The composition of the translations in one translations.

Solution:

We know that,
`(T\langle-3,4\rangle \circ T\langle 8,-7\rangle)(x,y)` means
`T\langle 8,-7\rangle (x,y)` is followed by
`T\langle -3,4\rangle (x,y)`.

In
`T\langle 8,-7\rangle (x,y)`,

`(x,y)\to (x+8,y-7)`

So,
`P(x,y)\to P'(x+8,y-7)`.

In
`T\langle -3,4\rangle (x,y)`,

`(x,y)\to (x-3,y+4)`

`P'(x+8,y-7)\to P''((x+8)-3,(y-7)+4)`

`P'(x+8,y-7)\to P''(x+5,y-3)`

Now, after composition of the translations

`P(x,y)\to P''(x+5,y-3)`

Here, rule of translation is

`(x,y)\to (x+5,y-3)`

Therefore, rule of translation is
`T\langle 5,-3\rangle (x,y)`.