Exercise 13 Page 564

We are given a segment with endpoints J and K and a point of rotation 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 a part of JK. The image of P is itself, so the rotated segment will also pass through the point P. From the fact that all the points are a part of the same segment, we can also conclude that m ∠ JPK is 180.

  

  The image of the point J is a point J' such that PJ'=PJ and m ∠ JPJ' =180. We have already found that m ∠ JPK =180, which means that J' lies on the same line as JK. To locate it, let's extend the given segment first.

  

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

  

  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 J'.

  

  We can follow the same reasoning to find K'. As we can see, the point P is the midpoint of JK. Therefore, the image of J is the point K and the image of K is the point J.