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 vertices

We need to map \(\Delta AQR\) to \(\Delta AKP\).
The correspondence of vertices is:

• \(A

ightarrow A\)

• \(Q

ightarrow K\)

• \(R

ightarrow P\)

Examine geometric markings

From the given diagram:

• \(AQ\) has a double tick mark, and \(AK\) has a double tick mark. Thus, \(AQ = AK\).
• \(AR\) has a single tick mark, and \(AP\) has a single tick mark. Thus, \(AR = AP\).
• \(\angle QAR\) has an angle arc, and \(\angle KAP\) has a matching angle arc. Thus, \(\angle QAR \cong \angle KAP\).

Determine the transformation

Since vertex \(A\) maps to itself (\(A
ightarrow A\)), the transformation must keep point \(A\) fixed.
Since \(AQ = AK\) and \(AR = AP\), and the angle \(\angle QAR\) is equal to \(\angle KAP\), rotating \(\Delta AQR\) about the fixed point \(A\) will rotate segment \(AQ\) onto segment \(AK\) and segment \(AR\) onto segment \(AP\).
Therefore, the rigid transformation is a rotation about point \(A\).

Answer:

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