Perform the two translations and, separately, the single translation. Does the knight end up at the correct location?
See solution.
Practice makes perfect
In the mentioned exercise, we first found two translations that describe two consecutive moves of a black knight during a chess game, and then rewrote them as a single translation.
To make sure both the two separate translations and the single translation we came up with make sense, we will test them on the knight and check if it takes us from the original position to the correct ending position. Let's start with the two separate translations!
Testing the Separate Translations
While solving the mentioned exercise, we came up with a translation for the first and second move of the knight.
First Move:(x,y) → (x+2,y-1)
Second Move: (x,y) → (x+1,y-2)
Now let's perform each translation and see whether they move the knight into the correct position. For the first translation, the x-value increases by 2, while the y-value decreases by 1. This tells us that the knight moves one tile down and two tiles right.
This is exactly the first move shown on the given diagram. Therefore, the first translation we have written is correct. Now let's check the second translation. Here, the x-value increases by 1 and the y-value decreases by 2. This tells us that the knight moves one tile right and two tiles down.
Once again, this is the same move as the one shown on the given diagram. This means that both translations are correct.
Testing the Single Translation
Now, we check whether the combined translation moves the knight from its original position to its ending position.
Single Translation: (x,y) → (x+3,y-3)
Here, the x-value increases by 3, while the y-value decreases by 3. This tells us that the knight moves right 3 tiles and down 3 tiles. Let's see if this puts the knight at the correct position.
As we can see, the single translation indeed moves the knight from the starting position to the ending position. Therefore, our translation is correct.
