Difference between revisions of "Rotation"
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We would first draw segment <math>AO</math>. Then, we would draw a new segment, <math>A'O</math> such that the angle formed is <math>60^{\circ}</math>, and <math>AO=A'O</math>. Do this for points <math>B</math> and <math>C</math>, to get the new triangle <math>A'B'C'</math> | We would first draw segment <math>AO</math>. Then, we would draw a new segment, <math>A'O</math> such that the angle formed is <math>60^{\circ}</math>, and <math>AO=A'O</math>. Do this for points <math>B</math> and <math>C</math>, to get the new triangle <math>A'B'C'</math> | ||
− | {{ | + | === Practice Problems === |
+ | *Isosceles <math>\triangle ABC</math> has a right angle at <math>C</math>. Point <math>P</math> is inside <math>\triangle ABC</math>, such that <math>PA=11</math>, <math>PB=7</math>, and <math>PC=6</math>. Legs <math>\overline{AC}</math> and <math>\overline{BC}</math> have length <math>s=\sqrt{a+b\sqrt{2}}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? | ||
+ | |||
+ | <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> | ||
+ | |||
+ | <math> | ||
+ | \mathrm{(A)}\ 85 | ||
+ | \qquad | ||
+ | \mathrm{(B)}\ 91 | ||
+ | \qquad | ||
+ | \mathrm{(C)}\ 108 | ||
+ | \qquad | ||
+ | \mathrm{(D)}\ 121 | ||
+ | \qquad | ||
+ | \mathrm{(E)}\ 127 | ||
+ | </math> | ||
+ | |||
+ | ([[2006 AMC 12B Problems/Problem 23|Source]]) |
Revision as of 22:17, 13 January 2021
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 clockwise around a point , also known as the center of rotation.
We would first draw segment . Then, we would draw a new segment, such that the angle formed is , and . Do this for points and , to get the new triangle
Practice Problems
- Isosceles has a right angle at . Point is inside , such that , , and . Legs and have length , where and are positive integers. What is ?
(Source)