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 the coordinate - change pattern

Given points \(A(-3,5)\) and \(A'(-3, - 5)\), \(B(2,8)\) and \(B'(2,-8)\), \(C(-4,-5)\) and \(C'(-4,5)\). We observe that the \(x\) - coordinates of the pre - image and the image points remain the same, while the \(y\) - coordinates change their signs.

Step2: Recall transformation rules

The transformation rule for reflection over the \(x\) - axis is \(r_{x - axis}(x,y)=(x,-y)\). In this rule, the \(x\) - value of a point stays the same and the \(y\) - value is negated, which is consistent with the given transformation of \(\triangle ABC\) to \(\triangle A'B'C'\).

Answer:

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