Question

is △nop congruent to △qrs? justify your answer using rigid motions. yes. a reflection can map △nop onto △qrs. yes. a rotation can map △nop onto △qrs. no. it is not possible to map △nop onto △qrs using only rigid motions.

Explanation:

Step1: Identify side - length and angle equalities

In \(\triangle NOP\) and \(\triangle QRS\), \(NO = 4\) in, \(NP=3\) in, \(OP = 5\) in, \(QR = 4\) in, \(QS = 3\) in, \(RS=5\) in. Also, \(\angle N=\angle Q = 90^{\circ}\), \(\angle O=\angle R\), \(\angle P=\angle S\).

Step2: Analyze rigid - motion

A rotation can map \(\triangle NOP\) onto \(\triangle QRS\). We can rotate \(\triangle NOP\) about a point such that the corresponding sides and angles match up with \(\triangle QRS\). A reflection would not work as the orientation of the triangles (in terms of the position of non - right angles) is not consistent with a reflection mapping.

Answer:

Yes. A rotation can map \(\triangle NOP\) onto \(\triangle QRS\).