Question

name the line of reflection used to map each preimage to its image.

Explanation:

Step1: Recall line - reflection rules

For a point \((x,y)\) reflected over the line \(y = k\), the \(y\) - coordinate changes as \(y'=2k - y\) and \(x\) - coordinate remains the same. For reflection over \(x = h\), the \(x\) - coordinate changes as \(x'=2h - x\) and \(y\) - coordinate remains the same. For reflection over \(y=x\), the coordinates swap \((x,y)\to(y,x)\) and for \(y =-x\), \((x,y)\to(-y,-x)\).

Step2: Analyze each pair of pre - image and image

For each figure, find the mid - points of the line segments connecting pre - image and image points. The line of reflection is the perpendicular bisector of these line segments.

1. For the first pair of figures (top - left):

• Observe the relative positions of points \(A\) and \(A'\), \(B\) and \(B'\). The line of reflection is \(x = 4\). We can check by finding the mid - points of \(AA'\) and \(BB'\) which lie on \(x = 4\) and the line \(x = 4\) is perpendicular to the line segments connecting pre - image and image points.

1. For the second pair of figures (top - right):

• By observing the positions of points \(L\) and \(L'\), \(K\) and \(K'\), etc., the line of reflection is \(y=-2\). The mid - points of the line segments connecting corresponding points lie on \(y=-2\) and \(y =-2\) is perpendicular to these line segments.

1. For the third pair of figures (middle - left):

• Analyzing the positions of points \(H\) and \(H'\), \(G\) and \(G'\), etc., the line of reflection is \(y=x\). We can verify this by noting that if a point \((x,y)\) is reflected over \(y = x\), it becomes \((y,x)\) and the mid - points of the line segments connecting pre - image and image points lie on \(y=x\) and \(y=x\) is perpendicular to these line segments.

1. For the fourth pair of figures (middle - right):

• Looking at the points \(E\) and \(E'\), \(F\) and \(F'\), etc., the line of reflection is \(x=-1\). The mid - points of the line segments connecting corresponding points lie on \(x=-1\) and \(x=-1\) is perpendicular to these line segments.

1. For the fifth pair of figures (bottom - left):

• Examining the points \(T\) and \(T'\), \(A\) and \(A'\), etc., the line of reflection is \(y=-x\). We can check that for a point \((x,y)\) reflected over \(y=-x\) gives \((-y,-x)\) and the mid - points of the line segments connecting pre - image and image points lie on \(y=-x\) and \(y=-x\) is perpendicular to these line segments.

1. For the sixth pair of figures (bottom - right):

• Analyzing the points \(W\) and \(W'\), \(X\) and \(X'\), etc., the line of reflection is \(y = 1\). The mid - points of the line segments connecting corresponding points lie on \(y = 1\) and \(y = 1\) is perpendicular to these line segments.

Answer:

1. \(x = 4\)
2. \(y=-2\)
3. \(y=x\)
4. \(x=-1\)
5. \(y=-x\)
6. \(y = 1\)