Chapter 14.4, Problem 14CE

To determine

To proof: Congruence invariant under half-turning mapping.

Answer to Problem 14CE

Yes, under half-turn mapping congruence is invariant.

Explanation of Solution

Given information :

Half-turning about a point $O$ through ${180}^{°}$ denoted by $H_O$ .

Concept used:

A rotation is a transformation suggested by rotating a paddle wheel. When wheel stops, new position of a paddle referred as the image of initial position.

As per the given problem

A rotation about point $O$ through ${180}^{°}$ denoted by $H_O$ . The diagram shown below $\Delta PQR$ and its image $\Delta P\text{'}Q\text{'}R\text{'}$ by $H_O$ . Notice that $O$ is mid-point of $\overline{PP\text{'}},\overline{QQ\text{'}}\&\overline{RR\text{'}}$

  

  $\begin{array}{l}PQ=P\text{'}Q\text{'}\\ QR=Q\text{'}R\text{'}\\ RP=R\text{'}P\text{'}\\ \angle P=\angle P\text{'}\\ \angle Q=\angle Q\text{'}\\ \angle R=\angle R\text{'}\end{array}$

  $\Delta PQR\widetilde{=}\Delta P\text{'}Q\text{'}R\text{'}$

Hence,

Under half-turn mapping congruence is invariant.