Question

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 $y = 4$?

(image of coordinate grid with triangle qrs: q(7, -3), r(10, -8), s(13, -4))

\\(\boldsymbol{\text{options:}}\\)
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: Apply Translation

For a translation vector \(\langle -12, 9
angle\), the rule is \((x, y) \to (x - 12, y + 9)\).

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

Step2: Apply Reflection over \(y = 4\)

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

• For \(Q'( -5, 6)\): \(y = 8 - 6 = 2\), so \(Q''( -5, 2)\). Wait, no, wait, earlier step had a mistake. Wait, no, let's recalculate. Wait, the translation: \(Q(7,-3)\) translated by \(\langle -12,9

angle\): \(7-12=-5\), \(-3 + 9 = 6\), so \(Q'(-5,6)\). Then reflection over \(y = 4\): the distance from \(y = 6\) to \(y = 4\) is \(6 - 4 = 2\), so we go down \(2\) from \(y = 4\), so \(4 - 2 = 2\)? Wait, no, the formula is \(y' = 2k - y\), where \(k = 4\), so \(y' = 8 - y\). So for \(Q'(-5,6)\), \(y' = 8 - 6 = 2\), so \(Q''(-5,2)\).

• For \(R'(-2,1)\): \(y' = 8 - 1 = 7\), so \(R''(-2,7)\).
• For \(S'(1,5)\): \(y' = 8 - 5 = 3\), so \(S''(1,3)\).

Wait, but the first step's translation gave \(Q'(-5,6)\), \(R'(-2,1)\), \(S'(1,5)\), then reflection over \(y = 4\) gives \(Q''(-5, 8 - 6)=(-5,2)\), \(R''(-2, 8 - 1)=(-2,7)\), \(S''(1, 8 - 5)= (1,3)\), which matches option C. Wait, but let's check again.

Wait, maybe I messed up the translation. Wait, the original points: \(Q(7,-3)\), \(R(10,-8)\), \(S(13,-4)\).

Translation vector \(\langle -12, 9
angle\): add \(-12\) to \(x\), add \(9\) to \(y\).

\(Q\): \(7 - 12 = -5\), \(-3 + 9 = 6\) → \(Q'(-5,6)\)

\(R\): \(10 - 12 = -2\), \(-8 + 9 = 1\) → \(R'(-2,1)\)

\(S\): \(13 - 12 = 1\), \(-4 + 9 = 5\) → \(S'(1,5)\)

Reflection over \(y = 4\): the formula is \((x, y) \to (x, 2*4 - y) = (x, 8 - y)\)

So \(Q''\): \(x = -5\), \(y = 8 - 6 = 2\) → \((-5,2)\)

\(R''\): \(x = -2\), \(y = 8 - 1 = 7\) → \((-2,7)\)

\(S''\): \(x = 1\), \(y = 8 - 5 = 3\) → \((1,3)\)

Which is option C: \(Q''(-5, 2)\), \(R''(-2, 7)\), \(S''(1, 3)\)

Answer:

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