Question

Can you find a shortcut to doing two translations

Answer 1

Answer: The explanation is given below.

Step-by-step explanation: We are given to find a shortcut to doing two translations.

Let us consider a point (x, y), which undergoes two translations one by one as follows :

(1) translation of a units to the right and b units upwards,

(2) another translation of c units to the left and d units downwards.

So, the translation rules will be

(x, y) ⇒ (x+a, y+b)

and

(x+a, y+b) ⇒ (x+a-c, y+b-d).

Instead of using two translations, we can directly use the following combined translation :

(x, y) ⇒ (x+a-c, y+b-d).

Thus, we can use one translation in place of two translations.

For example, suppose a point (2, 3) is first translated 5 units right and 2 units up and then translated 3 units left and 1 unit downwards.

Then, we can write the combined translation as follows :

(2, 3) ⇒ (2+5, 3+2) = (7, 5),

(7, 5) ⇒ (7-3,5-1) = (4, 4)

or directly,

(2, 3) ⇒ (2+5-3, 3+2-1) = (4, 4).

The translation of (2, 3) is shown in the attached figure below.

Hence explained.

Answer 2

A shortcut to doing two translations would be add both translations to get one translation IE:x+4 y+2 and x+2 y+4, are two translations, that are the same as x+6 y+6, which is only one translation. I hope the answer helps.