Difference between revisions of "2021 Fall AMC 12B Problems/Problem 10"

Latest revision as of 21:50, 10 July 2023

Problem

What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)$ is isosceles?

$\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380$

Solution

Let $A = (\cos 40^{\circ}, \sin 40^{\circ}), B = (\cos 60^{\circ}, \sin 60^{\circ}),$ and $C = (\cos t^{\circ}, \sin t^{\circ}).$ We apply casework to the legs of isosceles $\triangle ABC:$

$AB=AC$
Note that $A$ must be the midpoint of $\widehat{BC}.$ It follows that $C = (\cos 20^{\circ}, \sin 20^{\circ}),$ so $t=20.$

$BA=BC$
Note that $B$ must be the midpoint of $\widehat{AC}.$ It follows that $C = (\cos 80^{\circ}, \sin 80^{\circ}),$ so $t=80.$

$CA=CB$
Note that $C$ must be the midpoint of $\widehat{AB}.$ It follows that $C = (\cos 50^{\circ}, \sin 50^{\circ})$ or $C = (\cos 230^{\circ}, \sin 230^{\circ}),$ so $t=50$ or $t=230.$

Together, the sum of all such possible values of $t$ is $20+80+50+230=\boxed{\textbf{(E)} \: 380}.$

Remark

The following diagram shows all possible locations of $C:$

$[asy] /* Made by MRENTHUSIASM */ size(200); int xMin = -1; int xMax = 1; int yMin = -1; int yMax = 1; int numRays = 36; //Draws a polar grid that goes out to a number of circles //equal to big, with numRays specifying the number of rays: void polarGrid(int big, int numRays) { for (int i = 1; i < big+1; ++i) { draw(Circle((0,0),i), gray+linewidth(0.4)); } for (int i=0;i<numRays;++i) draw(rotate(i*360/numRays)*((-big,0)--(big,0)), gray+linewidth(0.4)); } polarGrid(xMax,numRays); draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),2*E); label("$y$",(0,yMax),2*N); pair A, B, C1, C2, C3, C4; A = dir(40); B = dir(60); C1 = dir(20); C2 = dir(80); C3 = dir(50); C4 = dir(230); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C_1$",C1,1.5*dir(C1),red+linewidth(4)); dot("$C_2$",C2,1.5*dir(C2),red+linewidth(4)); dot("$C_3$",C3,1.5*dir(C3),red+linewidth(4)); dot("$C_3$",C4,1.5*dir(C4),red+linewidth(4)); [/asy]$

~Steven Chen (www.professorchenedu.com) ~Wilhelm Z ~MRENTHUSIASM

Video Solution (Just 1 min!)

https://youtu.be/F_Hy5OWBC54

~Education, the Study of Everything

Video Solution by TheBeautyofMath

https://www.youtube.com/watch?v=4qgYrCYG-qw&t=1304

~IceMatrix

@@ Line 7: / Line 7: @@
 == Solution ==
-Denote <math>A = \left( \cos 40^\circ , \sin 40^\circ \right)</math>,
-<math>B = \left( \cos 60^\circ , \sin 60^\circ \right)</math>,
-and <math>C = \left( \cos t^\circ , \sin t^\circ \right)</math>.
-Case 1: <math>CA = CB</math>.
+Let <math>A = (\cos 40^{\circ}, \sin 40^{\circ}), B = (\cos 60^{\circ}, \sin 60^{\circ}),</math> and <math>C = (\cos t^{\circ}, \sin t^{\circ}).</math> We apply casework to the legs of isosceles <math>\triangle ABC:</math>
+<ol style="margin-left: 1.5em;">
+  <li><math>AB=AC</math><p>
+Note that <math>A</math> must be the midpoint of <math>\widehat{BC}.</math> It follows that <math>C = (\cos 20^{\circ}, \sin 20^{\circ}),</math> so <math>t=20.</math></li><p>
+  <li><math>BA=BC</math><p>
+Note that <math>B</math> must be the midpoint of <math>\widehat{AC}.</math> It follows that <math>C = (\cos 80^{\circ}, \sin 80^{\circ}),</math> so <math>t=80.</math></li><p>
+  <li><math>CA=CB</math><p>
+Note that <math>C</math> must be the midpoint of <math>\widehat{AB}.</math> It follows that <math>C = (\cos 50^{\circ}, \sin 50^{\circ})</math> or <math>C = (\cos 230^{\circ}, \sin 230^{\circ}),</math> so <math>t=50</math> or <math>t=230.</math>
+</li><p>
+</ol>
+Together, the sum of all such possible values of <math>t</math> is <math>20+80+50+230=\boxed{\textbf{(E)} \: 380}.</math>
-We have <math>t = 50</math> or <math>230</math>.
+<u><b>Remark</b></u>
-Case 2: <math>BA = BC</math>.
+The following diagram shows all possible locations of <math>C:</math>
-We have <math>t = 80</math>.
+<asy>
+/* Made by MRENTHUSIASM */
+size(200);
-Case 3: <math>AB = AC</math>.
+int xMin = -1;
+int xMax = 1;
+int yMin = -1;
+int yMax = 1;
+int numRays = 36;
-We have <math>t = 20</math>.
+//Draws a polar grid that goes out to a number of circles
+//equal to big, with numRays specifying the number of rays:
+void polarGrid(int big, int numRays)
+{
+  for (int i = 1; i < big+1; ++i)
+  {
+    draw(Circle((0,0),i), gray+linewidth(0.4));
+  }
+  for (int i=0;i<numRays;++i)
+  draw(rotate(i*360/numRays)*((-big,0)--(big,0)), gray+linewidth(0.4));
+}
-Therefore, the answer is <math>\boxed{\textbf{(E) }380}</math>.
+polarGrid(xMax,numRays);
+draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5));
+draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5));
+label("$x$",(xMax,0),2*E);
+label("$y$",(0,yMax),2*N);
+pair A, B, C1, C2, C3, C4;
+A = dir(40);
+B = dir(60);
+C1 = dir(20);
+C2 = dir(80);
+C3 = dir(50);
+C4 = dir(230);
+dot("$A$",A,1.5*dir(A),linewidth(4));
+dot("$B$",B,1.5*dir(B),linewidth(4));
+dot("$C_1$",C1,1.5*dir(C1),red+linewidth(4));
+dot("$C_2$",C2,1.5*dir(C2),red+linewidth(4));
+dot("$C_3$",C3,1.5*dir(C3),red+linewidth(4));
+dot("$C_3$",C4,1.5*dir(C4),red+linewidth(4));
+</asy>
 ~Steven Chen (www.professorchenedu.com) ~Wilhelm Z ~MRENTHUSIASM
+==Video Solution (Just 1 min!)==
+https://youtu.be/F_Hy5OWBC54
+<i>~Education, the Study of Everything</i>
+== Video Solution by TheBeautyofMath ==
+https://www.youtube.com/watch?v=4qgYrCYG-qw&t=1304
+~IceMatrix
+== See Also ==
 {{AMC12 box|year=2021 Fall|ab=B|num-a=11|num-b=9}}
 {{MAA Notice}}

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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