Question

for exercises 19 - 24, suppose m is the line with equation x = - 5, line n is the line with equation y = 1, line g is the line with equation y = x, and line h is the line with equation y = - 2. given a(9, - 3), b(6, 4), and c(- 1, - 5), what are the coordinates of the vertices of △abc for each reflection? see examples 2 and 3

1. r_x - axis
2. r_y - axis
3. r_m
4. r_n
5. r_g
6. r_h

Explanation:

Step1: Recall reflection rules

For reflection over x - axis, $(x,y)\to(x, - y)$; for y - axis, $(x,y)\to(-x,y)$; for line $x = a$, $(x,y)\to(2a - x,y)$; for line $y = b$, $(x,y)\to(x,2b - y)$; for line $y=x$, $(x,y)\to(y,x)$.

Step2: Reflection over x - axis ($R_{x - axis}$)

For point $A(9,-3)$, $A'(9,3)$; for $B(6,4)$, $B'(6, - 4)$; for $C(-1,-5)$, $C'(-1,5)$.

Step3: Reflection over y - axis ($R_{y - axis}$)

For point $A(9,-3)$, $A'(-9,-3)$; for $B(6,4)$, $B'(-6,4)$; for $C(-1,-5)$, $C'(1,-5)$.

Step4: Reflection over line $m:x=-5$ ($R_{m}$)

For point $A(9,-3)$, $x'=2\times(-5)-9=-10 - 9=-19$, $A'(-19,-3)$; for $B(6,4)$, $x'=2\times(-5)-6=-10 - 6=-16$, $B'(-16,4)$; for $C(-1,-5)$, $x'=2\times(-5)-(-1)=-10 + 1=-9$, $C'(-9,-5)$.

Step5: Reflection over line $n:y = 1$ ($R_{n}$)

For point $A(9,-3)$, $y'=2\times1-(-3)=2 + 3=5$, $A'(9,5)$; for $B(6,4)$, $y'=2\times1 - 4=-2$, $B'(6,-2)$; for $C(-1,-5)$, $y'=2\times1-(-5)=2 + 5=7$, $C'(-1,7)$.

Step6: Reflection over line $g:y=x$ ($R_{g}$)

For point $A(9,-3)$, $A'(-3,9)$; for $B(6,4)$, $B'(4,6)$; for $C(-1,-5)$, $C'(-5,-1)$.

Step7: Reflection over line $h:y=-2$ ($R_{h}$)

For point $A(9,-3)$, $y'=2\times(-2)-(-3)=-4 + 3=-1$, $A'(9,-1)$; for $B(6,4)$, $y'=2\times(-2)-4=-4 - 4=-8$, $B'(6,-8)$; for $C(-1,-5)$, $y'=2\times(-2)-(-5)=-4 + 5=1$, $C'(-1,1)$.

Answer:

1. $A'(9,3),B'(6, - 4),C'(-1,5)$
2. $A'(-9,-3),B'(-6,4),C'(1,-5)$
3. $A'(-19,-3),B'(-16,4),C'(-9,-5)$
4. $A'(9,5),B'(6,-2),C'(-1,7)$
5. $A'(-3,9),B'(4,6),C'(-5,-1)$
6. $A'(9,-1),B'(6,-8),C'(-1,1)$