Question

which set of points represents the image of \\( \triangle qrs \\) after a translation in the form
\\( (x,y) \
ightarrow(x - 12,y + 9) \\) followed by a reflection over the line \\( y = 4? \\)
a. \\( q(-5,6),r(-2,1),s(1,5) \\)
b. \\( q(7,11),r(10,16),s(13,12) \\)
c. \\( q(-5,2),r(-2,7),s(1,3) \\)
d. \\( q(13,2),r(10,7),s(7,3) \\)

Explanation:

Step1: Find the coordinates after translation

For a point \((x,y)\), the translation \((x,y)\to(x - 12,y + 9)\) is applied.

• For \(Q(7,-3)\): \(x=7,y=-3\), after translation \(x_1=7-12=-5,y_1=-3 + 9=6\), so \(Q'( - 5,6)\)
• For \(R(10,-8)\): \(x = 10,y=-8\), after translation \(x_1=10-12=-2,y_1=-8 + 9=1\), so \(R'(-2,1)\)
• For \(S(13,-4)\): \(x = 13,y=-4\), after translation \(x_1=13-12 = 1,y_1=-4 + 9=5\), so \(S'(1,5)\)

Step2: Find the coordinates after reflection over \(y = 4\)

The formula for reflection over the line \(y = k\) is \((x,y)\to(x,2k - y)\). Here \(k = 4\), so \((x,y)\to(x,8 - y)\)

• For \(Q'(-5,6)\): \(x=-5,y = 6\), after reflection \(x_2=-5,y_2=8 - 6=2\), so \(Q''(-5,2)\)
• For \(R'(-2,1)\): \(x=-2,y = 1\), after reflection \(x_2=-2,y_2=8 - 1=7\), so \(R''(-2,7)\)
• For \(S'(1,5)\): \(x = 1,y = 5\), after reflection \(x_2=1,y_2=8 - 5=3\), so \(S''(1,3)\)

Answer:

C. \(Q''(-5,2),R''(-2,7),S''(1,3)\)