Question

suppose line m is the line with the equation ( x = - 5 ), line n is the line with the equation ( y = 1 ), line g is the line with the equation ( y = x ), and line h is the line with the equation ( y = - 2 ). given ( a(9,-3) ), ( b(6,4) ), and ( c(-1,-5) ), what are the coordinates of the vertices of ( \triangle abc ) for the reflection ( r_g )?
a ( (-3,9) ) (type an ordered pair.)
b ( (4,6) ) (type an ordered pair.)
c ( square ) (type an ordered pair.)

Explanation:

Step1: Reflection over \(y = x\)

For a point \((x,y)\) reflected over \(y=x\), the rule is \((x,y)\to(y,x)\).
For \(A(9, - 3)\), after reflection over \(y = x\), \(A'\) is \((-3,9)\).
For \(B(6,4)\), after reflection over \(y = x\), \(B'\) is \((4,6)\).

Step2: Reflection over \(y=-2\)

The formula for reflecting a point \((x,y)\) over the line \(y = k\) is \((x,y)\to(x,2k - y)\). Here \(k=-2\).
For \(C(-1,-5)\), using the formula \(x=-1\), \(y = 2\times(-2)-(-5)=-4 + 5=1\). So \(C'=(-1,1)\)

Answer:

A. \((-3,9)\), B. \((4,6)\), C. \((-1,1)\)