Rotation

A rotation of a planar figure is a transformation that preserves area and angles, but not orientation. The resulting figure is congruent to the first.

Suppose we wish to rotate triangle $ABC$ $60^{\circ}$ clockwise around a point $O$ , also known as the center of rotation.

We would first draw segment $AO$ . Then, we would draw a new segment, $A'O$ such that the angle formed is $60^{\circ}$ , and $AO=A'O$ . Do this for points $B$ and $C$ , to get the new triangle $A'B'C'$

Practice Problems

Isosceles $\triangle ABC$ has a right angle at $C$ . Point $P$ is inside $\triangle ABC$ , such that $PA=11$ , $PB=7$ , and $PC=6$ . Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}}$ , where $a$ and $b$ are positive integers. What is $a+b$ ?

$[asy] pathpen = linewidth(0.7); pen f = fontsize(10); size(5cm); pair B = (0,sqrt(85+42*sqrt(2))); pair A = (B.y,0); pair C = (0,0); pair P = IP(arc(B,7,180,360),arc(C,6,0,90)); D(A--B--C--cycle); D(P--A); D(P--B); D(P--C); MP("A",D(A),plain.E,f); MP("B",D(B),plain.N,f); MP("C",D(C),plain.SW,f); MP("P",D(P),plain.NE,f); [/asy]$

$\mathrm{(A)}\ 85 \qquad \mathrm{(B)}\ 91 \qquad \mathrm{(C)}\ 108 \qquad \mathrm{(D)}\ 121 \qquad \mathrm{(E)}\ 127$

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Rotation

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