Question

what is the rule used to transform $\triangle abc$ to its image?
$a(-3,5), b(2,8), c(-4,-5)$ and $a(-3,-5), b(2,-8), c(-4,5)$
a. $r_m(x,y)=(-y,-x)$, where the equation of line $m$ is $y = -x$
b. $r_n(x,y)=(y,x)$, where the equation of line $n$ is $y = -x$
c. $r_{y - axis}(x,y)=(-x,y)$
d. $r_{x - axis}(x,y)=(x,-y)$

Explanation:

Step1: Analyze coordinate - changes

For point \(A(-3,5)\) and its image \(A'(-3, - 5)\), \(x\) - coordinate remains the same (\(-3\)), and \(y\) - coordinate changes sign (\(5\) becomes \(-5\)). For point \(B(2,8)\) and its image \(B'(2,-8)\), \(x\) - coordinate is \(2\) (unchanged) and \(y\) - coordinate changes sign (\(8\) becomes \(-8\)). For point \(C(-4,-5)\) and its image \(C'(-4,5)\), \(x\) - coordinate is \(-4\) (unchanged) and \(y\) - coordinate changes sign (\(-5\) becomes \(5\)).

Step2: Recall transformation rules

The rule for reflection over the \(x\) - axis is \(r_{x - axis}(x,y)=(x,-y)\), which matches the observed coordinate - changes.

Answer:

D. \(r_{x - axis}(x,y)=(x,-y)\)