Question

the image of the point $(-5,4)$ under a translation is $(-3,-1)$. find the coordinates of the image of the point $(-2,9)$ under the same translation.

Explanation:

Step1: Find the translation rule

Let the translation rule be \((x,y)\to(x + a,y + b)\).
For the point \((-5,4)\) and its image \((-3,-1)\), we have:
\(-5+a=-3\), so \(a=-3+ 5=2\);
\(4 + b=-1\), so \(b=-1 - 4=-5\).
The translation rule is \((x,y)\to(x + 2,y-5)\).

Step2: Apply the translation rule to the point \((-2,9)\)

For the point \((-2,9)\), when \(x=-2\) and \(y = 9\):
\(x+2=-2 + 2=0\);
\(y-5=9-5 = 4\).
So the image of the point \((-2,9)\) under the translation is \((0,4)\).

Answer:

$(0,4)$