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Difference between revisions of "2019 AMC 8 Problems/Problem 24"

(→‎Solution 1)
(→‎Solution 1)
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==Solution 1==
 
==Solution 1==
 
Draw <math>X</math> so that <math>XD</math> is parallel to <math>BC</math>. That makes triangles <math>BEF</math> and <math>EXD</math> congruent since <math>BE</math>=<math>ED</math>. <math>FC</math>=3<math>XD</math> so <math>BC</math>=4<math>BC</math>. Since <math>AF</math>=3<math>EF</math>( <math>XE</math>=<math>EF</math> and <math>AX</math>=1/3<math>AF</math>, so <math>XE</math>=<math>EF</math>=1/3<math>AF</math>), the altitude of triangle <math>BEF</math> is equal to 1/3 of the altitude of <math>ABC</math>. The area of <math>ABC</math> is 360, so the area of <math>BEF</math>=1/3*1/4*360=<math>\boxed{(B) 30}</math>~heeeeeeeheeeee
 
Draw <math>X</math> so that <math>XD</math> is parallel to <math>BC</math>. That makes triangles <math>BEF</math> and <math>EXD</math> congruent since <math>BE</math>=<math>ED</math>. <math>FC</math>=3<math>XD</math> so <math>BC</math>=4<math>BC</math>. Since <math>AF</math>=3<math>EF</math>( <math>XE</math>=<math>EF</math> and <math>AX</math>=1/3<math>AF</math>, so <math>XE</math>=<math>EF</math>=1/3<math>AF</math>), the altitude of triangle <math>BEF</math> is equal to 1/3 of the altitude of <math>ABC</math>. The area of <math>ABC</math> is 360, so the area of <math>BEF</math>=1/3*1/4*360=<math>\boxed{(B) 30}</math>~heeeeeeeheeeee
 +
 +
==Solution 2 (Mass Points)==
 +
<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 +
import graph; size(7cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -6.28, xmax = 6.28, ymin = -5.49, ymax = 5.73;  /* image dimensions */
 +
pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451);
 +
/* draw figures */
 +
draw((0.28,2.39)--(-2.8,-1.17), linewidth(2) + wrwrwr);
 +
draw((-2.8,-1.17)--(3.78,-1.05), linewidth(2) + wrwrwr);
 +
draw((3.78,-1.05)--(0.28,2.39), linewidth(2) + wrwrwr);
 +
draw((-2.8,-1.17)--(1.2887445398528459,1.3985482236874887), linewidth(2) + wrwrwr);
 +
draw((0.28,2.39)--(-0.7199623188673492,-1.1320661821070033), linewidth(2) + wrwrwr);
 +
draw(circle((-0.1,2.93), 0.46818799642878495), linewidth(2) + wrwrwr);
 +
draw(circle((-0.1,2.93), 0.46818799642878495), linewidth(2) + wrwrwr);
 +
draw(circle((4.48,-1.28), 0.46818799642878506), linewidth(2) + wrwrwr);
 +
draw(circle((1.98,1.56), 0.46818799642878495), linewidth(2) + wrwrwr);
 +
draw(circle((-3.36,-1.62), 0.46818799642878517), linewidth(2) + wrwrwr);
 +
draw(circle((0.16,0.14), 0.46818799642878495), linewidth(2) + wrwrwr);
 +
draw(circle((-0.74,-1.81), 0.46818799642878495), linewidth(2) + wrwrwr);
 +
/* dots and labels */
 +
dot((0.28,2.39),dotstyle);
 +
label("$A$", (0.36,2.59), NE * labelscalefactor);
 +
dot((-2.8,-1.17),dotstyle);
 +
label("$B$", (-2.72,-0.97), NE * labelscalefactor);
 +
dot((3.78,-1.05),dotstyle);
 +
label("$C$", (3.86,-0.85), NE * labelscalefactor);
 +
dot((1.2887445398528459,1.3985482236874887),dotstyle);
 +
label("$D$", (1.36,1.59), NE * labelscalefactor);
 +
dot((-0.7199623188673492,-1.1320661821070033),dotstyle);
 +
label("$F$", (-0.64,-0.93), NE * labelscalefactor);
 +
dot((-0.2815567696989588,0.41208536204620183),linewidth(4pt) + dotstyle);
 +
label("$E$", (-0.2,0.57), NE * labelscalefactor);
 +
label("2", (-0.18,2.81), NE * labelscalefactor,wrwrwr);
 +
label("1", (4.4,-1.39), NE * labelscalefactor,wrwrwr);
 +
label("3", (1.9,1.45), NE * labelscalefactor,wrwrwr);
 +
label("3", (-3.44,-1.73), NE * labelscalefactor,wrwrwr);
 +
label("6", (0.08,0.03), NE * labelscalefactor,wrwrwr);
 +
label("4", (-0.82,-1.93), NE * labelscalefactor,wrwrwr);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
/* end of picture */
 +
</asy>
 +
 +
First, when we see the problem, we see ratios, and we see that this triangle basically has no special properties (right, has medians, etc.) and this screams mass points at us.
 +
 +
First, we assign a mass of <math>2</math> to point <math>A</math>. We figure out that <math>C</math> has a mass of <math>1</math> since <math>2\times1 = 1\times2</math>. Then, by adding <math>1+2 = 3</math>, we get that point <math>D</math> has a mass of 3. By equality, point <math>B</math> has a mass of 3 also.
 +
 +
Now, we add <math>3+3 = 6</math> for point <math>E</math> and <math>3+1 = 4</math> for point <math>F</math>.
 +
 +
Now, <math>BF</math> is a common base for triangles <math>ABF</math> and <math>EBF</math>, so we figure out that the ratios of the areas is the ratios of the heights which is <math>\frac{AE}{EF} = 2:1</math>. So, <math>EBF</math>'s area is one third the area of <math>ABF</math>, and we know the area of <math>ABF</math> is <math>\frac{1}{4}</math> the area of <math>ABC</math> since they have the same heights but different bases.
 +
 +
So we get the area of <math>EBF</math> as <math>\frac{1}{3}\times\frac{1}{4}\times360 = \boxed{B}30)</math>
 +
-Brudder
 +
Note: We can also find the ratios of the areas using the reciprocal of the product of the mass points of <math>EBF</math> over the product of the mass points of <math>ABC</math> which is <math>\frac{2\times3\times1}{3\times6\times4}\times360</math> which also yields <math>\boxed{B}</math>
 +
-Brudder
  
 
==See Also==
 
==See Also==

Revision as of 20:37, 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

Draw $X$ so that $XD$ is parallel to $BC$. That makes triangles $BEF$ and $EXD$ congruent since $BE$=$ED$. $FC$=3$XD$ so $BC$=4$BC$. Since $AF$=3$EF$( $XE$=$EF$ and $AX$=1/3$AF$, so $XE$=$EF$=1/3$AF$), the altitude of triangle $BEF$ is equal to 1/3 of the altitude of $ABC$. The area of $ABC$ is 360, so the area of $BEF$=1/3*1/4*360=$\boxed{(B) 30}$~heeeeeeeheeeee

Solution 2 (Mass Points)

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(7cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.28, xmax = 6.28, ymin = -5.49, ymax = 5.73; /* image dimensions */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); /* draw figures */ draw((0.28,2.39)--(-2.8,-1.17), linewidth(2) + wrwrwr); draw((-2.8,-1.17)--(3.78,-1.05), linewidth(2) + wrwrwr); draw((3.78,-1.05)--(0.28,2.39), linewidth(2) + wrwrwr); draw((-2.8,-1.17)--(1.2887445398528459,1.3985482236874887), linewidth(2) + wrwrwr); draw((0.28,2.39)--(-0.7199623188673492,-1.1320661821070033), linewidth(2) + wrwrwr); draw(circle((-0.1,2.93), 0.46818799642878495), linewidth(2) + wrwrwr); draw(circle((-0.1,2.93), 0.46818799642878495), linewidth(2) + wrwrwr); draw(circle((4.48,-1.28), 0.46818799642878506), linewidth(2) + wrwrwr); draw(circle((1.98,1.56), 0.46818799642878495), linewidth(2) + wrwrwr); draw(circle((-3.36,-1.62), 0.46818799642878517), linewidth(2) + wrwrwr); draw(circle((0.16,0.14), 0.46818799642878495), linewidth(2) + wrwrwr); draw(circle((-0.74,-1.81), 0.46818799642878495), linewidth(2) + wrwrwr); /* dots and labels */ dot((0.28,2.39),dotstyle); label("$A$", (0.36,2.59), NE * labelscalefactor); dot((-2.8,-1.17),dotstyle); label("$B$", (-2.72,-0.97), NE * labelscalefactor); dot((3.78,-1.05),dotstyle); label("$C$", (3.86,-0.85), NE * labelscalefactor); dot((1.2887445398528459,1.3985482236874887),dotstyle); label("$D$", (1.36,1.59), NE * labelscalefactor); dot((-0.7199623188673492,-1.1320661821070033),dotstyle); label("$F$", (-0.64,-0.93), NE * labelscalefactor); dot((-0.2815567696989588,0.41208536204620183),linewidth(4pt) + dotstyle); label("$E$", (-0.2,0.57), NE * labelscalefactor); label("2", (-0.18,2.81), NE * labelscalefactor,wrwrwr); label("1", (4.4,-1.39), NE * labelscalefactor,wrwrwr); label("3", (1.9,1.45), NE * labelscalefactor,wrwrwr); label("3", (-3.44,-1.73), NE * labelscalefactor,wrwrwr); label("6", (0.08,0.03), NE * labelscalefactor,wrwrwr); label("4", (-0.82,-1.93), NE * labelscalefactor,wrwrwr); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

First, when we see the problem, we see ratios, and we see that this triangle basically has no special properties (right, has medians, etc.) and this screams mass points at us.

First, we assign a mass of $2$ to point $A$. We figure out that $C$ has a mass of $1$ since $2\times1 = 1\times2$. Then, by adding $1+2 = 3$, we get that point $D$ has a mass of 3. By equality, point $B$ has a mass of 3 also.

Now, we add $3+3 = 6$ for point $E$ and $3+1 = 4$ for point $F$.

Now, $BF$ is a common base for triangles $ABF$ and $EBF$, so we figure out that the ratios of the areas is the ratios of the heights which is $\frac{AE}{EF} = 2:1$. So, $EBF$'s area is one third the area of $ABF$, and we know the area of $ABF$ is $\frac{1}{4}$ the area of $ABC$ since they have the same heights but different bases.

So we get the area of $EBF$ as $\frac{1}{3}\times\frac{1}{4}\times360 = \boxed{B}30)$ -Brudder Note: We can also find the ratios of the areas using the reciprocal of the product of the mass points of $EBF$ over the product of the mass points of $ABC$ which is $\frac{2\times3\times1}{3\times6\times4}\times360$ which also yields $\boxed{B}$ -Brudder

See Also

2019 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

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