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Difference between revisions of "2019 AIME II Problems/Problem 1"

m (→‎Solution)
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-Solution by the Math Wizard, Number Magician of the Second Order, Head of the Council of the Geometers
 
-Solution by the Math Wizard, Number Magician of the Second Order, Head of the Council of the Geometers
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==Solution 2==
 +
 +
Using the diagram in Solution 1, let <math>E</math> be the intersection of <math>BD</math> and <math>AC</math>. We can see that angle <math>C</math> is in both
 +
<math>\triangle BCE</math> and <math>\triangle ABC</math>. Since <math>\triangle BCE</math> and <math>\triangle ADE</math> are congruent by AAS, we can then state <math>AE=BE</math> and <math>DE=CE</math>. It follows that <math>BE=AE</math> and <math>CE=17-BE</math>. We can now state that the area of <math>\triangle ABE</math> is the area of <math>\triangle ABC-</math> the area of <math>\triangle BCE</math>. Using Heron's formula, we compute the area of <math>\triangle ABC=36</math>. Using the Law of Cosines on angle <math>C</math>, we obtain
 +
 +
<cmath>9^2=17^2+10^2-2(17)(10)cosC</cmath>
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<cmath>-308=-340cosC</cmath>
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<cmath>cosC=\frac{308}{340}</cmath>
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(For convenience, we're not going to simplify.)
 +
 +
Applying the Law of Cosines on <math>\triangle BCE</math> yields
 +
<cmath>BE^2=10^2+(17-BE)^2-2(10)(17-BE)cosC</cmath>
 +
<cmath>BE^2=389-34BE+BE^2-20(17-BE)(\frac{308}{340})</cmath>
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<cmath>0=389-34BE-(340-20BE)(\frac{308}{340})</cmath>
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<cmath>0=389-34BE+\frac{308BE}{17}</cmath>
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<cmath>0=81-\frac{270BE}{17}</cmath>
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<cmath>81=\frac{270BE}{17}</cmath>
 +
<cmath>BE=\frac{51}{10}</cmath>
 +
This means <math>CE=17-BE=17-\frac{51}{10}=\frac{119}{10}</math>. Next, apply Heron's formula to get the area of <math>\triangle BCE</math>, which equals <math>\frac{126}{5}</math> after simplifying. Subtracting the area of <math>\triangle BCE</math> from the area of <math>\triangle ABC</math> yields the area of <math>\triangle ABE</math>, which is <math>\frac{54}{5}</math>, giving us our answer, which is <math>54+5=\boxed{059}.</math>
 +
-Solution by flobszemathguy
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2019|n=II|before=First Problem|num-a=2}}
 
{{AIME box|year=2019|n=II|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:21, 22 March 2019

Problem

Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

[asy] unitsize(10); pair A = (0,0); pair B = (9,0); pair C = (15,8); pair D = (-6,8); draw(A--B--C--cycle); draw(B--D--A); label("$A$",A,dir(-120)); label("$B$",B,dir(-60)); label("$C$",C,dir(60)); label("$D$",D,dir(120)); label("$9$",(A+B)/2,dir(-90)); label("$10$",(D+A)/2,dir(-150)); label("$10$",(C+B)/2,dir(-30)); label("$17$",(D+B)/2,dir(60)); label("$17$",(A+C)/2,dir(120)); draw(D--(-6,0)--A,dotted); label("$8$",(D+(-6,0))/2,dir(180)); label("$6$",(A+(-6,0))/2,dir(-90)); [/asy] - Diagram by Brendanb4321


Extend $AB$ to form a right triangle with legs $6$ and $8$ such that $AD$ is the hypotenuse and connect the points $CD$ so that you have a rectangle. The base $CD$ of the rectangle will be $9+6+6=21$. Now, let $E$ be the intersection of $BD$ and $AC$. This means that $\triangle ABE$ and $\triangle DCE$ are with ratio $\frac{21}{9}=\frac73$. Set up a proportion, knowing that the two heights add up to 8. We will let $y$ be the height from $E$ to $DC$, and $x$ be the height of $\triangle ABE$. \[\frac{7}{3}=\frac{y}{x}\] \[\frac{7}{3}=\frac{8-x}{x}\] \[7x=24-3x\] \[10x=24\] \[x=\frac{12}{5}\]

This means that the area is $A=\tfrac{1}{2}(9)(\tfrac{12}{5})=\tfrac{54}{5}$. This gets us $54+5=\boxed{059}.$

-Solution by the Math Wizard, Number Magician of the Second Order, Head of the Council of the Geometers

Solution 2

Using the diagram in Solution 1, let $E$ be the intersection of $BD$ and $AC$. We can see that angle $C$ is in both $\triangle BCE$ and $\triangle ABC$. Since $\triangle BCE$ and $\triangle ADE$ are congruent by AAS, we can then state $AE=BE$ and $DE=CE$. It follows that $BE=AE$ and $CE=17-BE$. We can now state that the area of $\triangle ABE$ is the area of $\triangle ABC-$ the area of $\triangle BCE$. Using Heron's formula, we compute the area of $\triangle ABC=36$. Using the Law of Cosines on angle $C$, we obtain

\[9^2=17^2+10^2-2(17)(10)cosC\] \[-308=-340cosC\] \[cosC=\frac{308}{340}\] (For convenience, we're not going to simplify.)

Applying the Law of Cosines on $\triangle BCE$ yields \[BE^2=10^2+(17-BE)^2-2(10)(17-BE)cosC\] \[BE^2=389-34BE+BE^2-20(17-BE)(\frac{308}{340})\] \[0=389-34BE-(340-20BE)(\frac{308}{340})\] \[0=389-34BE+\frac{308BE}{17}\] \[0=81-\frac{270BE}{17}\] \[81=\frac{270BE}{17}\] \[BE=\frac{51}{10}\] This means $CE=17-BE=17-\frac{51}{10}=\frac{119}{10}$. Next, apply Heron's formula to get the area of $\triangle BCE$, which equals $\frac{126}{5}$ after simplifying. Subtracting the area of $\triangle BCE$ from the area of $\triangle ABC$ yields the area of $\triangle ABE$, which is $\frac{54}{5}$, giving us our answer, which is $54+5=\boxed{059}.$ -Solution by flobszemathguy

See Also

2019 AIME II (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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