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2019 AMC 8 Problems/Problem 24: Difference between revisions

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==Problem==
 
 
 
==Problem 24==
In triangle <math>ABC</math>, point <math>D</math> divides side <math>\overline{AC}</math> s that <math>AD</math>:<math>DC=1</math>:<math>2</math>. Let <math>E</math> be the midpoint of <math>\overline{BD}</math> and left <math>F</math> be the point of intersection of line <math>BC</math> and line <math>AE</math>. Given that the area of <math>\triangle ABC</math> is <math>360</math>, what is the area of <math>\triangle EBF</math>?
 
<asy>
<asy>
unitsize(2cm);
unitsize(2cm);
Line 23: Line 28:
label("$F$",FF,S);
label("$F$",FF,S);
</asy>
</asy>
<math>\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40</math>


==Solution 1==
==Solution 1==
==Solution 2==
 
{{AMC8 box|year=2019|num-b=23|num-a=25}}
 
==See Also==
{{AMC8 box|year=2019|num-b=19|num-a=21}}
 
{{MAA Notice}}

Revision as of 16:37, 20 November 2019


Problem 24

In triangle $ABC$, point $D$ divides side $\overline{AC}$ s that $AD$:$DC=1$:$2$. Let $E$ be the midpoint of $\overline{BD}$ and left $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?

[asy] unitsize(2cm); pair A,B,C,DD,EE,FF; B = (0,0); C = (3,0); A = (1.2,1.7); DD = (2/3)*A+(1/3)*C; EE = (B+DD)/2; FF = intersectionpoint(B--C,A--A+2*(EE-A)); draw(A--B--C--cycle); draw(A--FF); draw(B--DD);dot(A); label("$A$",A,N); dot(B); label("$B$", B,SW);dot(C); label("$C$",C,SE); dot(DD); label("$D$",DD,NE); dot(EE); label("$E$",EE,NW); dot(FF); label("$F$",FF,S); [/asy]


$\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40$

Solution 1

See Also

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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 AJHSME/AMC 8 Problems and Solutions

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

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