Difference between revisions of "Rotation"

Latest revision as of 22:22, 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 $ABC$ $60^{\circ}$ 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 $a+b$ ?

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

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

(Source)

Suppose that $\triangle{ABC}$ is an equilateral triangle of side length $s$ , with the property that there is a unique point $P$ inside the triangle such that $AP=1$ , $BP=\sqrt{3}$ , and $CP=2$ . What is $s$ ?

$\textbf{(A) } 1+\sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5+\sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$

(Source)

Three concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \tfrac{b}{c} \sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d.$

(Source)

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Difference between revisions of "Rotation"

Latest revision as of 22:22, 13 January 2021

Practice Problems

@@ Line 5: / Line 5: @@
 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>
-{{stub}}
+=== 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]])
+*Suppose that <math>\triangle{ABC}</math> is an equilateral triangle of side length <math>s</math>, with the property that there is a unique point <math>P</math> inside the triangle such that <math>AP=1</math>, <math>BP=\sqrt{3}</math>, and <math>CP=2</math>. What is <math>s</math>?
+<math>\textbf{(A) } 1+\sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5+\sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}</math>
+([[2020 AMC 12A Problems/Problem 24|Source]])
+*Three concentric circles have radii <math>3,</math> <math>4,</math> and <math>5.</math> An equilateral triangle with one vertex on each circle has side length <math>s.</math> The largest possible area of the triangle can be written as <math>a + \tfrac{b}{c} \sqrt{d},</math> where <math>a,</math> <math>b,</math> <math>c,</math> and <math>d</math> are positive integers, <math>b</math> and <math>c</math> are relatively prime, and <math>d</math> is not divisible by the square of any prime. Find <math>a+b+c+d.</math>
+([[2012 AIME I Problems/Problem 13|Source]])