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2020 AMC 12A Problems/Problem 24: Difference between revisions

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giving us that <math>s = \boxed{\textbf{(B) } \sqrt{7}}</math>. ~ciceronii
giving us that <math>s = \boxed{\textbf{(B) } \sqrt{7}}</math>. ~ciceronii
==Solution 2 (Super Bash)==
<asy>
draw((0,0)--(4,5.65)--(8,0)--cycle);
label("A", (4,5.65), N, p = fontsize(10pt));
label("C", (8,0), SE, p = fontsize(10pt));
label("B", (0,0), SW, p = fontsize(10pt));
label("P", (3.5,3.5), NW, p = fontsize(10pt));
draw((0,0)--(3.5,3.5));
label("$\sqrt{3}$",(0,0)--(3.5,3.5), SE);
draw((8,0)--(3.5,3.5));
label("2",(8,0)--(3.5,3.5), SW);
draw((4,5.65)--(3.5,3.5));
label("1",(4,5.65)--(3.5,3.5), E);
</asy>
We begin by rotating <math>\triangle{ABC}</math> by <math>60^{\circ}</math> about <math>A</math>, such that in <math>\triangle{A'B'C'}</math>, <math>B' = C</math>. We see that
<math>\triangle{APP'}</math> is equilateral with side length <math>1</math>, meaning that <math>\angle APP' = 60^{\circ}</math>. We also see that <math>\triangle{CPP'}</math> is a <math>30-60-90</math> right triangle, meaning that <math>\angle CPP'= 60^{\circ}</math>. Thus, by adding the two together, we see that <math>\angle APC = 120^{\circ}</math>. We can now use the law of cosines as following:
<cmath>s^2 = (AP)^2 + (CP)^2 - 2(AP)(CP)\cos{\angle{APC}}</cmath>
<cmath>s^2 = 1 + 4 - 2(1)(2)\cos{120^{\circ}}</cmath>
<cmath>s^2 = 5 - 4(-\frac{1}{2})</cmath>
<cmath>s = \sqrt{5 + 2}</cmath>
giving us that <math>s = \boxed{\textbf{(B) } \sqrt{7}}</math>. ~ciceronii
==Video Solution==
==Video Solution==
https://www.youtube.com/watch?v=mUW4zcrRL54
https://www.youtube.com/watch?v=mUW4zcrRL54

Revision as of 12:21, 3 February 2020

Problem 24

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 $s$?

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

Solution

[asy] draw((0,0)--(4,5.65)--(8,0)--cycle); label("A", (4,5.65), N, p = fontsize(10pt)); label("C", (8,0), SE, p = fontsize(10pt)); label("B", (0,0), SW, p = fontsize(10pt)); label("P", (3.5,3.5), NW, p = fontsize(10pt)); draw((0,0)--(3.5,3.5)); label("$\sqrt{3}$",(0,0)--(3.5,3.5), SE); draw((8,0)--(3.5,3.5)); label("2",(8,0)--(3.5,3.5), SW); draw((4,5.65)--(3.5,3.5)); label("1",(4,5.65)--(3.5,3.5), E); draw((8,0)--(4,5.65)--(11,5.65)--cycle); label("$P'$", (6,4.75), NE, p = fontsize(10pt)); draw((4,5.65)--(6,4.75)); label("1",(4,5.65)--(6,4.75), S); draw((8,0)--(6,4.75)); label("$\sqrt{3}$",(8,0)--(6,4.75), E); draw((3.5,3.5)--(6,4.75)); label("1", (3.5,3.5)--(6,4.75), SE); [/asy]

We begin by rotating $\triangle{ABC}$ by $60^{\circ}$ about $A$, such that in $\triangle{A'B'C'}$, $B' = C$. We see that $\triangle{APP'}$ is equilateral with side length $1$, meaning that $\angle APP' = 60^{\circ}$. We also see that $\triangle{CPP'}$ is a $30-60-90$ right triangle, meaning that $\angle CPP'= 60^{\circ}$. Thus, by adding the two together, we see that $\angle APC = 120^{\circ}$. We can now use the law of cosines as following:

\[s^2 = (AP)^2 + (CP)^2 - 2(AP)(CP)\cos{\angle{APC}}\] \[s^2 = 1 + 4 - 2(1)(2)\cos{120^{\circ}}\] \[s^2 = 5 - 4(-\frac{1}{2})\] \[s = \sqrt{5 + 2}\]

giving us that $s = \boxed{\textbf{(B) } \sqrt{7}}$. ~ciceronii

Solution 2 (Super Bash)

[asy] draw((0,0)--(4,5.65)--(8,0)--cycle); label("A", (4,5.65), N, p = fontsize(10pt)); label("C", (8,0), SE, p = fontsize(10pt)); label("B", (0,0), SW, p = fontsize(10pt)); label("P", (3.5,3.5), NW, p = fontsize(10pt)); draw((0,0)--(3.5,3.5)); label("$\sqrt{3}$",(0,0)--(3.5,3.5), SE); draw((8,0)--(3.5,3.5)); label("2",(8,0)--(3.5,3.5), SW); draw((4,5.65)--(3.5,3.5)); label("1",(4,5.65)--(3.5,3.5), E); [/asy]

We begin by rotating $\triangle{ABC}$ by $60^{\circ}$ about $A$, such that in $\triangle{A'B'C'}$, $B' = C$. We see that $\triangle{APP'}$ is equilateral with side length $1$, meaning that $\angle APP' = 60^{\circ}$. We also see that $\triangle{CPP'}$ is a $30-60-90$ right triangle, meaning that $\angle CPP'= 60^{\circ}$. Thus, by adding the two together, we see that $\angle APC = 120^{\circ}$. We can now use the law of cosines as following:

\[s^2 = (AP)^2 + (CP)^2 - 2(AP)(CP)\cos{\angle{APC}}\] \[s^2 = 1 + 4 - 2(1)(2)\cos{120^{\circ}}\] \[s^2 = 5 - 4(-\frac{1}{2})\] \[s = \sqrt{5 + 2}\]

giving us that $s = \boxed{\textbf{(B) } \sqrt{7}}$. ~ciceronii

Video Solution

https://www.youtube.com/watch?v=mUW4zcrRL54

See Also

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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
All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America.

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