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.
