Question

in the diagram at the right, qrst is a rectangle with rs = 2ts.
a. copy the diagram. then sketch $r_{\overleftrightarrow{qt}}$(qrst).
b. what figure results from the reflection? use properties of reflections
to justify your solution.

a. which diagram below shows $r_{\overleftrightarrow{qt}}$(qrst)?
\\(\bigcirc\\) a.
\\(\bigcirc\\) b.
\\(\bigcirc\\) c.

diagrams for a, b, c are shown with respective labels and structures as described in the original image

Explanation:

Step1: Recall Reflection Properties

A reflection over a line (here, \(\overleftrightarrow{QT}\)) maps each point to a point such that the line is the perpendicular bisector of the segment joining the original point and its image. For rectangle \(QRST\), \(QT\) is a vertical side (from the diagram). Reflecting over \(\overleftrightarrow{QT}\) will mirror the rectangle across \(QT\).

Step2: Analyze Side Lengths

Given \(RS = 2TS\), so \(RS\) (horizontal side) is twice \(TS\) (vertical side). When reflecting over \(QT\) (vertical line), the horizontal distance from each point to \(QT\) will be mirrored. For point \(R\), its image \(R'\) should be equidistant from \(QT\) on the left, and similarly for \(S\) (image \(S'\)) and \(T\) (image \(T'\), but \(T\) is on \(QT\), so \(T' = T\), \(Q\) is on \(QT\), so \(Q' = Q\)).

Step3: Evaluate Options

- **Option A**: The reflected figure (left of \(QT\)) has the same horizontal length as \(QRST\) (right of \(QT\)), forming a larger rectangle. Since \(RS = 2TS\), reflecting \(RS\) (length \(2TS\)) over \(QT\) would add a segment of length \(RS\) to the left, making the total horizontal length \(RS + RS = 2RS = 4TS\)? Wait, no—wait, original \(RS\) is horizontal, length \(2TS\). Reflecting over \(QT\) (vertical) would mirror the horizontal side: the distance from \(R\) to \(QT\) is \(RS\) (since \(QRST\) has \(QR\) vertical, \(RS\) horizontal). Wait, maybe better to look at the diagram. In the original, \(QRST\) is a tall rectangle? No, \(RS = 2TS\), so \(RS\) (horizontal) is longer than \(TS\) (vertical). Wait, the original diagram: \(Q\) and \(T\) are on the vertical line, \(R\) and \(S\) are to the right. Reflecting over \(QT\) (vertical) would put \(R'\) and \(S'\) to the left of \(QT\), with \(R'S' = RS\) (since reflection preserves length) and \(TS' = TS\) (since \(T\) is on \(QT\)). Wait, Option A shows the reflected figure (left) with the same horizontal length as the original (right), so the combined figure is a rectangle with width \(2 \times RS\) (since original \(RS\) is right of \(QT\), reflected \(RS'\) is left of \(QT\)), and height \(TS\). Wait, but original \(RS = 2TS\), so \(RS\) (horizontal) is \(2TS\) (vertical). So reflecting over \(QT\) (vertical) would make the left side (reflected) have \(R'S' = RS = 2TS\), same as right. So the total width is \(RS + R'S' = 2RS = 4TS\)? No, wait, maybe I misread: \(RS = 2TS\), so \(RS\) (horizontal) is twice \(TS\) (vertical). So \(TS\) is vertical, length \(x\), \(RS\) is horizontal, length \(2x\). Then \(QRST\) has vertical sides \(QT\) (length \(TS = x\)) and \(QR\) (same as \(TS\)), horizontal sides \(RS\) (length \(2x\)) and \(QT\) is vertical. Wait, no, rectangle: opposite sides equal. So \(QT = RS\)? No, no—rectangle \(QRST\): \(QR\) and \(TS\) are vertical, \(QT\) and \(RS\) are horizontal? Wait, no, labels: \(Q\), \(R\), \(S\), \(T\) in order, so \(QR\) is top, \(RS\) is right, \(ST\) is bottom, \(TQ\) is left. So \(QR\) and \(ST\) are vertical, \(RS\) and \(TQ\) are horizontal? Wait, no, that can't be. Wait, standard rectangle labeling: \(Q\) to \(R\) to \(S\) to \(T\) to \(Q\). So \(QR\) and \(TS\) are vertical, \(RS\) and \(QT\) are horizontal. So \(QT\) is horizontal? Wait, the diagram has a vertical arrow, so \(QT\) is vertical (since \(Q\) and \(T\) are on the vertical line). So \(QT\) is vertical, \(RS\) is horizontal (from \(R\) to \(S\), right side), \(TS\) is vertical (from \(T\) to \(S\), bottom side). So \(RS\) (horizontal) length \(= 2 \times TS\) (vertical) length. So \(RS\) is horizontal, length \(2TS\), \(TS\) is vertical, length \(x\), so \(RS = 2x\). Reflecting over vertical line \(QT\) (which contains \(Q\) and \(T\)): each point \(P\) has image \(P'\) such that \(QT\) is perpendicular bisector of \(PP'\). So \(R\) (right of \(QT\)) has image \(R'\) left of \(QT\), with \(QR' = QR\) (vertical) and \(R'S' = RS\) (horizontal). So the reflected figure (left of \(QT\)) will have \(R'S' = RS = 2x\), same as \(RS\) (right of \(QT\)). So the combined figure (Option A) is a rectangle with width \(RS + R'S' = 2x + 2x = 4x\)? No, wait, original \(RS\) is from \(QT\) to \(R\), length \(2x\). Reflecting over \(QT\) would put \(R'\) at distance \(2x\) left of \(QT\), so the segment \(R'S'\) is length \(2x\), same as \(RS\). So the figure in A is two rectangles (left and right of \(QT\)) each with horizontal length \(2x\), vertical length \(x\), forming a larger rectangle with horizontal length \(4x\), vertical length \(x\)? Wait, no, maybe the original \(RS\) is length \(2TS\), so \(TS\) is vertical, \(RS\) is horizontal, so \(QRST\) has height \(TS\) and width \(RS = 2TS\). Reflecting over the vertical line \(QT\) (which is the left side of \(QRST\)? Wait, no—wait, in the original diagram, \(Q\) and \(T\) are on the vertical line (the axis of reflection), \(R\) and \(S\) are to the right of \(QT\). So \(QT\) is the left edge of the rectangle \(QRST\)? No, because \(Q\) is top, \(T\) is bottom, \(R\) is top-right, \(S\) is bottom-right. So \(QT\) is the left vertical side, \(QR\) is the top horizontal side? Wait, no, that would make \(QRST\) a rectangle with left side \(QT\), top side \(QR\), right side \(RS\), bottom side \(ST\). So \(QT\) and \(RS\) are vertical, \(QR\) and \(ST\) are horizontal. Then \(RS = 2TS\): \(RS\) is vertical length, \(TS\) is horizontal length? Wait, now I'm confused. Let's re-express: let’s assign coordinates. Let \(Q = (0, h)\), \(T = (0, 0)\), so \(QT\) is the vertical line \(x = 0\). Then \(R = (w, h)\), \(S = (w, 0)\), so \(QRST\) has vertices at \((0, h)\), \((w, h)\), \((w, 0)\), \((0, 0)\). Then \(RS\) is the segment from \((w, h)\) to \((w, 0)\), length \(h\) (vertical). \(TS\) is the segment from \((0, 0)\) to \((w, 0)\), length \(w\) (horizontal). Given \(RS = 2TS\), so \(h = 2w\). So \(RS\) (vertical) length \(2w\), \(TS\) (horizontal) length \(w\). Now, reflecting over \(\overleftrightarrow{QT}\) (which is \(x = 0\)): the reflection of a point \((x, y)\) over \(x = 0\) is \((-x, y)\). So \(R = (w, h)\) reflects to \(R' = (-w, h)\), \(S = (w, 0)\) reflects to \(S' = (-w, 0)\), \(Q = (0, h)\) reflects to \(Q' = (0, h)\), \(T = (0, 0)\) reflects to \(T' = (0, 0)\). So the reflected figure \(R'Q'T'S'\) has vertices \((-w, h)\), \((0, h)\), \((0, 0)\), \((-w, 0)\), which is a rectangle with width \(w\) (from \(x = -w\) to \(x = 0\)) and height \(h = 2w\) (from \(y = 0\) to \(y = h\)). The original figure \(QRST\) has width \(w\) (from \(x = 0\) to \(x = w\)) and height \(2w\). So together, the two rectangles (original and reflected) form a larger rectangle with width \(w + w = 2w\) and height \(2w\)? Wait, no—original width is \(w\) (from \(0\) to \(w\)), reflected width is \(w\) (from \(-w\) to \(0\)), so total width \(2w\), height \(2w\)? But \(h = 2w\), so height is \(2w\), width is \(2w\), so a square? No, wait, original \(RS = 2TS\): \(RS\) is vertical length \(h = 2w\), \(TS\) is horizontal length \(w\), so \(h = 2w\) implies \(RS = 2TS\) (since \(TS = w\), \(RS = 2w\)). Then reflecting over \(x = 0\) (QT) gives a rectangle on the left with width \(w\) (same as original) and height \(2w\) (same as original). So the combined figure (original + reflected) has width \(2w\) and height \(2w\)? Wait, no, the original is from \(x=0\) to \(x=w\), reflected is from \(x=-w\) to \(x=0\), so total width \(2w\), height \(2w\), so a square? But in Option A, the diagram shows the reflected figure (left) with the same dimensions as the original (right), forming a larger rectangle. Let's check the options:

- **Option B**: The reflected figure (left) is a smaller rectangle, same height as original but narrower. But since reflection preserves length, \(R'S'\) should be equal to \(RS\) (vertical length), but \(RS\) is vertical, so reflecting over vertical line \(QT\) would not change the vertical length. Wait, no—if \(QT\) is vertical, reflecting over it would mirror horizontal distances, not vertical. So vertical sides (QR, TS) remain vertical, horizontal sides (RS, QT) remain horizontal. Wait, I think I messed up the coordinate system. Let's re-express:

Let me look at the original diagram (top right): \(Q\) is top, \(T\) is bottom, both on the vertical arrow (line \(QT\)), \(R\) is top-right, \(S\) is bottom-right. So \(QT\) is vertical, \(QR\) is horizontal (from \(Q\) to \(R\)), \(RS\) is vertical (from \(R\) to \(S\)), \(ST\) is horizontal (from \(S\) to \(T\)). So \(QRST\) has horizontal sides \(QR\) and \(ST\), vertical sides \(RS\) and \(QT\). Then \(RS = 2TS\): \(RS\) is vertical length, \(TS\) is horizontal length (from \(T\) to \(S\), which is the same as \(QR\), horizontal length). So \(RS\) (vertical) = \(2 \times TS\) (horizontal). So \(QRST\) is a tall rectangle: vertical side \(RS\) is twice the horizontal side \(TS\). Now, reflecting over \(QT\) (vertical line): each horizontal point (R, S) will be mirrored across \(QT\). So \(R\) is to the right of \(QT\), so its image \(R'\) is to the left of \(QT\), same horizontal distance from \(QT\) as \(R\) is. \(S\) is to the right of \(QT\), image \(S'\) is to the left, same distance. \(Q\) and \(T\) are on \(QT\), so their images are themselves.

So \(QR\) is horizontal (from \(Q\) to \(R\), rightward), length \(TS\) (since \(TS\) is horizontal from \(T\) to \(S\), same as \(QR\)). \(RS\) is vertical (from \(R\) to \(S\)), length \(2 \times TS\). So reflecting \(R\) over \(QT\) gives \(R'\) such that \(QR' = QR\) (horizontal leftward), and \(R'S' = RS\) (vertical downward, same as \(RS\) downward). Wait, no—if \(QR\) is horizontal (right), then \(QR'\) should be horizontal (left), same length. \(RS\) is vertical (down), so \(R'S'\) is vertical (down), same length. So the reflected figure \(QR'S'T\) (wait, no, the reflection is \(R_{\overleftrightarrow{QT}}(QRST)\), so the image of \(QRST\) is \(Q'R'S'T'\), where \(Q' = Q\), \(T' = T\), \(R' = \text{reflection of } R\), \(S' = \text{reflection of } S\).

So \(R\) is right of \(QT\), so \(R'\) is left of \(QT\), same horizontal distance from \(QT\) as \(R\) is. \(S\) is right of \(QT\), so \(S'\) is left of \(QT\), same horizontal distance. So the horizontal segment \(RS\) (right of \(QT\)) has length \(L\) (let \(TS = x\), so \(RS = 2x\), so horizontal length \(TS = x\), vertical length \(RS = 2x\)? Wait, now I'm really confused. Let's look at the options:

- **Option A**: The diagram shows the original rectangle (right of QT) and a reflected rectangle (left of QT) with the same horizontal length, forming a larger rectangle. This matches the reflection over QT: mirroring the horizontal side (RS) over QT, so the left rectangle has the same dimensions as the right, making a larger rectangle.

- **Option B**: The reflected rectangle is narrower, which would mean the horizontal length is shorter, but reflection preserves length, so this is wrong.

- **Option C**: The reflected figure is rotated or misplaced, not a mirror image over QT.

So the correct diagram is Option A.

Answer:

A. The diagram with the reflected rectangle (left of \( \overleftrightarrow{QT} \)) having the same dimensions as \( QRST \) (right of \( \overleftrightarrow{QT} \)), forming a larger rectangle.