Question

the image of the point $(6,-4)$ under a translation is $(7,-7)$. find the coordinates of the image of the point $(1,-5)$ under the same translation.

Explanation:

Step1: Find the translation rule

The translation rule \((x,y)\to(x + a,y + b)\).
For the point \((6,-4)\) and its image \((7,-7)\), we have \(6 + a=7\) and \(-4 + b=-7\).
From \(6 + a=7\), we get \(a=7 - 6=1\).
From \(-4 + b=-7\), we get \(b=-7+ 4=-3\). So the translation rule is \((x,y)\to(x + 1,y-3)\).

Step2: Apply the translation rule to the point \((1,-5)\)

For the point \((1,-5)\), when \(x = 1\) and \(y=-5\).
The \(x\)-coordinate of the image is \(x+1=1 + 1=2\).
The \(y\)-coordinate of the image is \(y - 3=-5-3=-8\).

Answer:

\((2,-8)\)