Exercise 1 Page 564

We are given a triangle with vertices $A,$ $B,$ and $C,$ and a point of rotation $P.$

We want to draw $r_{\left(70^{\circ },P\right)}\left(△ABC\right).$ This means that we want to rotate $△ABC$ by $70^{\circ }$ about point $P.$ To do so, we will follow four steps.

1. Draw $\overline{PC}$ and construct a $70^{\circ }$ angle with vertex $P$ and side $\overline{PC}.$
2. Construct $\overline{P{C}^{\prime }}$ such that $\overline{P{C}^{\prime }}$ lies on a side of the angle drawn in the previous step and $\overline{P{C}^{\prime }}\cong \overline{PC}.$
3. Locate $A^{\prime }$ and $B^{\prime }$ in a similar manner.
4. Connect $A^{\prime },$ $B^{\prime },$ and $C^{\prime }$ to draw $△A^{\prime }{B}^{\prime }{C}^{\prime }.$

Let's do it!

Step $1$

Let's draw $\overline{PC}.$

We will use a protractor to construct a $70^{\circ }$ angle with vertex $P$ and side $\overline{PC}.$ We start by placing the center of the protractor on $P,$ making sure that the flat part is on $\overline{PC}.$

Notice that the inner measuring scale has $0^{\circ }$ on $\overline{PC}.$ Therefore, we will use the inner measuring scale. Moreover, recall that if it is not specified we measure the angle in the counterclockwise direction.

Finally, to construct the angle, we remove the protractor and draw a ray from $P$ that passes through the mark we have just drawn.

Note that for this step we could have chosen point $A$ or point $B$ instead of $C.$ We arbitrarily chose point $C$ for simplicity.

Step $2$

We will now locate $C^{\prime },$ which is the image of $C$ after the rotation. To do so we will use a compass. We will start by placing the sharp spike of the compass at $P,$ and the leg with the pencil at $C.$

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 $C^{\prime }.$

Step $3$

We can repeat Step $1$ and $2$ to find $A^{\prime }.$

Then we can repeat the same steps to locate $B^{\prime }.$

Step $4$

Finally, to draw $△A^{\prime }{B}^{\prime }{C}^{\prime }$ — which is the image of $△ABC$ after a rotation of $70^{\circ }$ about $P$ — we will connect the obtained vertices.