Question

suppose m is the line with equation y = -4 and \\( \triangle a ^ { prime } b ^ { prime } c \\) is mapped to \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\) by applying the glide reflection \\( t _ { ( 3,0 ) } cdot r _ { m } \\). what are the coordinates of \\( \triangle a ^ { prime prime } b ^ { prime prime } c ^ { prime prime } \\)?
a \\( a ^ { prime prime } ( - 1, - 3 ), b ^ { prime prime } ( - 2, - 6 ), c ^ { prime prime } ( 1, - 7 ) \\)
b \\( a ^ { prime prime } ( - 7, - 3 ), b ^ { prime prime } ( - 8, - 6 ), c ^ { prime prime } ( - 5, - 7 ) \\)
c \\( a ^ { prime } ( - 2, - 5 ), b ^ { prime } ( 0, - 2 ), c ^ { prime } ( - 3, - 1 ) \\)
d \\( a ^ { prime } ( - 8, - 3 ), b ^ { prime } ( - 6,4 ), c ^ { prime } ( - 9,0 ) \\)

Explanation:

Step1: Find the coordinates of \(A'\), \(B'\), \(C'\)

From the graph, \(A'(-2,-5)\), \(B'(-6,-2)\), \(C'(-3,-1)\)

Step2: Apply reflection \(R_m\) (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 \(A'(-2,-5)\): \(x=-2\), \(y = - 5\), then \(A''(-2,-8+5)=(-2,-3)\)
• For \(B'(-6,-2)\): \(x=-6\), \(y=-2\), then \(B''(-6,-8 + 2)=(-6,-6)\)
• For \(C'(-3,-1)\): \(x=-3\), \(y=-1\), then \(C''(-3,-8+1)=(-3,-7)\)

Step3: Apply translation \(T_{(x,y)}\) (translation by \((-5,3)\))

The formula for translation \((x,y)\to(x + a,y + b)\), here \(a=-5\), \(b = 3\)

• For \(A''(-2,-3)\): \((-2-5,-3 + 3)=(-7,-3)\)
• For \(B''(-6,-6)\): \((-6-5,-6 + 3)=(-8,-6)\)
• For \(C''(-3,-7)\): \((-3-5,-7 + 3)=(-5,-7)\)

Answer:

B. \(A''(-7,-3)\), \(B''(-8,-6)\), \(C''(-5,-7)\)