Question

reflections and translations

1. triangle abc is transformed to its image abc, shown in the coordinate plane.

kira says, \triangle abc was reflected over the y - axis to its image abc\
londan says, \triangle abc was translated 8 units to the right to its image abc\
who is correct? explain your reasoning below.

Explanation:

Step1: Find coordinates of points

Let's assume coordinates of \(A(-2,1)\), \(B(-6,1)\), \(C(-4,3)\) and \(A'(6,1)\), \(B'(2,1)\), \(C'(4,3)\)

Step2: Check reflection over y - axis

Reflection over \(y\) - axis: \((x,y)\to(-x,y)\). If \(A(-2,1)\) is reflected over \(y\) - axis, it would be \((2,1)\) not \((6,1)\)

Step3: Check translation

Translation rule: \((x,y)\to(x + h,y + k)\). For \(A(-2,1)\) to \(A'(6,1)\): \(x\) - coordinate: \(-2+h = 6\Rightarrow h = 8\), \(y\) - coordinate: \(1 + k=1\Rightarrow k = 0\). Similarly for \(B(-6,1)\) to \(B'(2,1)\): \(-6+h = 2\Rightarrow h = 8\), \(y\) - coordinate: \(1 + k = 1\Rightarrow k = 0\) and for \(C(-4,3)\) to \(C'(4,3)\): \(-4+h=4\Rightarrow h = 8\), \(y\) - coordinate: \(3 + k = 3\Rightarrow k = 0\)

Answer:

London is correct. The translation rule \((x,y)\to(x + 8,y)\) maps \(A(-2,1)\) to \(A'(6,1)\), \(B(-6,1)\) to \(B'(2,1)\) and \(C(-4,3)\) to \(C'(4,3)\) while reflection over \(y\) - axis does not map the points correctly.