Question

14 which set of points represents the image of \\( \triangle qrs \\) after a translation along \\( \langle - 12,9 \
angle \\) followed by a reflection in the line \\( x = 4 \\) ? * (10 points) \\( q ^ { \prime } ( 7,11 ) , r ^ { \prime } ( 10,16 ) , s ^ { \prime } ( 13,12 ) \\) \\( q ^ { \prime } ( - 5,6 ) , r ^ { \prime } ( - 2,1 ) , s ^ { \prime } ( 1,5 ) \\) \\( q ^ { \prime } ( - 5,2 ) , r ^ { \prime } ( - 2,7 ) , s ^ { \prime } ( 1,3 ) \\) \\( q ^ { \prime } ( 13,6 ) , r ^ { \prime } ( 10,1 ) , s ^ { \prime } ( 7,5 \\)

Explanation:

Step1: Translation

Translation along \(\langle - 12,9
angle\) means \(x\) - coordinate \(x'=x - 12\) and \(y\) - coordinate \(y'=y + 9\).
For \(Q(7,-3)\): \(x'=7-12=-5\), \(y'=-3 + 9=6\), so \(Q_1(-5,6)\)
For \(R(10,-8)\): \(x'=10-12=-2\), \(y'=-8 + 9=1\), so \(R_1(-2,1)\)
For \(S(13,-4)\): \(x'=13-12 = 1\), \(y'=-4 + 9=5\), so \(S_1(1,5)\)

Step2: Reflection in the line \(x = 4\)

The formula for reflection of a point \((x,y)\) in the line \(x=a\) is \((2a - x,y)\). Here \(a = 4\), so the new \(x\) - coordinate \(x''=2\times4-x\) and \(y''=y\)
For \(Q_1(-5,6)\): \(x''=2\times4-(-5)=8 + 5=13\), \(y''=6\), so \(Q'(13,6)\)
For \(R_1(-2,1)\): \(x''=2\times4-(-2)=8 + 2=10\), \(y''=1\), so \(R'(10,1)\)
For \(S_1(1,5)\): \(x''=2\times4-1=8 - 1=7\), \(y''=5\), so \(S'(7,5)\)

Answer:

\(Q'(13,6),R'(10,1),S'(7,5)\) (the fourth option)