Question

describe the transformation or sequence of transformations that map each of the preimages to the images listed below.

1. abcd → ghcd
2. mnop → abcd
3. rqop → lmjk
4. gfed → rqop

Explanation:

Let's solve problem 6: \(ABCD \to GHCD\)

Step 1: Identify Common Vertices

Notice that points \(C\) and \(D\) are common to both \(ABCD\) and \(GHCD\). So we focus on transforming \(A\) and \(B\) to \(G\) and \(H\).

Step 2: Determine Transformation Type

Looking at the diagram (assuming horizontal reflection or translation/rotation). From the positions, \(ABCD\) to \(GHCD\) can be a reflection over a vertical line (or a horizontal flip? Wait, actually, since \(CD\) is common, let's check the horizontal axis. Wait, the figure has a horizontal line (the base) and vertical segments. Alternatively, a reflection over the vertical line through \(CD\)? Wait, no, let's see the coordinates (even without coordinates, the shape: \(ABCD\) and \(GHCD\) share \(CD\), so \(AB\) is transformed to \(GH\). So the transformation is a reflection over the vertical line that is the perpendicular bisector of \(AG\) (or \(BH\))? Wait, maybe a reflection over the vertical line through \(D\) (or \(C\))? Alternatively, a horizontal reflection? Wait, actually, in the diagram, \(ABCD\) and \(GHCD\) are symmetric with respect to a vertical line (the line through \(D\) and \(C\) maybe? No, \(D\) and \(C\) are on the same vertical segment? Wait, \(D\) is below \(C\)? Wait, the diagram shows \(A---B---C\) on the top, \(D\) below \(C\), \(G\) to the right of \(D\), \(H\) to the right of \(B\)? Wait, maybe a reflection over the vertical line (the y-axis-like line in the diagram). Alternatively, a translation? No, because \(C\) and \(D\) are fixed? Wait, no, \(C\) and \(D\) are common, so the transformation is a reflection over the vertical line that passes through \(CD\) (since \(CD\) is vertical? Wait, \(CD\) is a vertical segment (from \(C\) down to \(D\)). So \(ABCD\) has \(AB\) horizontal, \(BC\) vertical? Wait, no, the diagram: \(A\) is left, \(B\) above, \(C\) right of \(B\), \(D\) below \(C\), then \(G\) is right of \(D\), \(H\) is right of \(B\), \(I\) above \(G\), etc. So \(ABCD\) to \(GHCD\): \(A\) maps to \(G\), \(B\) maps to \(H\), \(C\) maps to \(C\), \(D\) maps to \(D\). So the transformation is a reflection over the vertical line that is the perpendicular bisector of \(AG\) (or \(BH\)). Alternatively, a horizontal reflection? Wait, maybe a reflection over the vertical line through \(D\) (or \(C\)). Let's think: if we reflect \(ABCD\) over the vertical line passing through \(CD\) (since \(CD\) is vertical), then \(A\) (left of \(CD\)) would map to \(G\) (right of \(CD\)), \(B\) (left of \(CD\)) would map to \(H\) (right of \(CD\)), \(C\) and \(D\) are on the line, so they map to themselves. So the transformation is a reflection over the vertical line containing segment \(CD\).

Answer:

The transformation from \(ABCD\) to \(GHCD\) is a reflection over the vertical line that contains the segment \(CD\) (since points \(C\) and \(D\) are invariant, and points \(A\) and \(B\) are reflected across this vertical line to map to \(G\) and \(H\) respectively).

(For other problems, follow similar steps: identify common points, determine the type of transformation (reflection, translation, rotation, dilation) by analyzing the position and shape of preimage and image.)