Question

directions: graph and label each figure and its image under a reflection in the given line. give the coordinates of the image. 1. square bcde with vertices b(-6,7), c(-2,6), d(-3,2), and e(-7,3): y - axis 2. triangle fgh with vertices f(1,8), g(5,7), and h(2,3): y = x 3. trapezoid jklm with vertices j(-4,3), k(-2,7), l(2,7), and m(3,3): y = 1 4. rhombus wxyz with vertices w(1,5), x(6,3), y(1,1), and z(-4,3): x - axis 5. triangle pqr with vertices p(-2,3), q(2,4), and r(1,-1): x = -3 6. rectangle mnop with vertices m(-7,1), n(-4,1), o(-4,-4), and p(-7,-4): y = -x

Explanation:

Step1: Recall reflection rules

• Reflection over the $y$-axis: $(x,y)\to(-x,y)$
• Reflection over $y = x$: $(x,y)\to(y,x)$
• Reflection over $y = 1$: $(x,y)\to(x,2 - y)$
• Reflection over the $x$-axis: $(x,y)\to(x,-y)$
• Reflection over $x=-3$: $(x,y)\to(-6 - x,y)$
• Reflection over $y=-x$: $(x,y)\to(-y,-x)$

Step2: Calculate new coordinates for Square BCDE (reflection over $y$-axis)

• For $B(-6,7)$: $B'=(6,7)$
• For $C(-2,6)$: $C'=(2,6)$
• For $D(-3,2)$: $D'=(3,2)$
• For $E(-7,3)$: $E'=(7,3)$

Step3: Calculate new coordinates for Triangle FGH (reflection over $y = x$)

• For $F(1,8)$: $F'=(8,1)$
• For $G(5,7)$: $G'=(7,5)$
• For $H(2,3)$: $H'=(3,2)$

Step4: Calculate new coordinates for Trapezoid JKLM (reflection over $y = 1$)

• For $J(-4,3)$: $J'(-4,-1)$
• For $K(-2,7)$: $K'(-2,-5)$
• For $L(2,7)$: $L'(2,-5)$
• For $M(3,3)$: $M'(3,-1)$

Step5: Calculate new coordinates for Rhombus WXYZ (reflection over $x$-axis)

• For $W(1,5)$: $W'(1,-5)$
• For $X(6,3)$: $X'(6,-3)$
• For $Y(1,1)$: $Y'(1,-1)$
• For $Z(-4,3)$: $Z'(-4,-3)$

Step6: Calculate new coordinates for Triangle PQR (reflection over $x=-3$)

• For $P(-2,3)$: $P'(-4,3)$
• For $Q(2,4)$: $Q'(-8,4)$
• For $R(1,-1)$: $R'(-7,-1)$

Step7: Calculate new coordinates for Rectangle MNOP (reflection over $y=-x$)

• For $M(-7,1)$: $M'(-1,7)$
• For $N(-4,1)$: $N'(-1,4)$
• For $O(-4,-4)$: $O'(4,4)$
• For $P(-7,-4)$: $P'(4,7)$

Answer:

1. $B'(6,7), C'(2,6), D'(3,2), E'(7,3)$
2. $F'(8,1), G'(7,5), H'(3,2)$
3. $J'(-4,-1), K'(-2,-5), L'(2,-5), M'(3,-1)$
4. $W'(1,-5), X'(6,-3), Y'(1,-1), Z'(-4,-3)$
5. $P'(-4,3), Q'(-8,4), R'(-7,-1)$
6. $M'(-1,7), N'(-1,4), O'(4,4), P'(4,7)$