Exercise 16 Page 564

We are given a segment with endpoints J and K and a point of rotation P. Notice that J=P.

We want to rotate JK by 180^(∘) about point P. Recall the two properties of a rotation of 180 ^(∘) about a point P.

• The image of P is itself (that is, P'=P).
• For any other point V, PV'=PV and m ∠ VPV'=180.

  In this case, the point P is the endpoint of JK. The image of P is itself. Therefore, the rotated segment will also pass through the point P. Moreover, because J=P, the image of J is itself.

  

  The image of the point K is a point K' such that PK'=PK and m ∠ KPK' =180. To locate it, let's extend the given segment first.

  

  Now, we will use a compass to find K'. We start by placing the sharp spike of the compass at P and the leg with the pencil at K.

  

  Without modifying the amplitude of the compass, we will keep the sharp spike at P. Then we will draw an arc intersecting the ray we drew in the previous step. This point of intersection is K'.

  

  Finally, to draw J'K' — which is the image of JK after a rotation of 180^(∘) about P — we will connect the obtained vertices.