Question

which rigid transformation would map \\(\delta aqr\\) to \\(\delta akp\\)?

a rotation about point a
a reflection across the line containing \\(\overline{ar}\\)
a reflection across the line containing \\(\overline{aq}\\)
a rotation about point r

Explanation:

Analyze corresponding parts of the triangles

Using the Congruence Mapping knowledge point
The mapping is \(\Delta AQR \to \Delta AKP\).
This means:

• Vertex \(A\) maps to vertex \(A\).
• Vertex \(Q\) maps to vertex \(K\).
• Vertex \(R\) maps to vertex \(P\).

Determine the type of transformation

Using the Rigid Transformations knowledge point

• Since vertex \(A\) maps to itself, point \(A\) is the center of the transformation.
• The orientation of the vertices \(A \to Q \to R\) is counterclockwise, while \(A \to K \to P\) is clockwise.
• A change in orientation indicates a reflection rather than a rotation.
• Thus, we reflect across a line passing through the fixed point \(A\).

Identify the line of reflection

Using the Rigid Transformations knowledge point

• The line of reflection must bisect the angle between corresponding points.
• Specifically, the line must bisect \(\angle QAK\) and \(\angle RAP\).
• From the diagram, the tick marks show \(AR = AP\) and \(AQ = AK\).
• The angle markings show \(\angle QAR \cong \angle KAP\).
• Reflecting across the line containing \(\overline{AR}\) maps \(\overline{AR}\) to itself (which is incorrect since \(R\) must map to \(P\)).
• Reflecting across the line containing \(\overline{AQ}\) maps \(\overline{AQ}\) to itself (which is incorrect since \(Q\) must map to \(K\)).
• Therefore, we must rotate about the fixed point \(A\) to map the segments: segment \(AQ\) rotates to segment \(AK\), and segment \(AR\) rotates to segment \(AP\).
• Let's re-verify the orientation: \(A \to Q \to R\) is counterclockwise. \(A \to K \to P\) is also counterclockwise.
• Since orientation is preserved, the transformation is a rotation about the common vertex \(A\).

Answer:

• (A) a rotation about point A (Correct answer)
• (B) a reflection across the line containing \(\overline{AR}\)
• (C) a reflection across the line containing \(\overline{AQ}\)
• (D) a rotation about point R