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

(Solution 7)
(Solution 7)
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== Solution 7 ==
 
== Solution 7 ==
A[∆ABD]: A[∆DBC] :: 1:2 :: 120:240
+
<math>A[\Delta ABD]: A[\Delta DBC] :: 1:2 :: 120:240</math>
  
  
A[∆ABE] = A[∆AED] = 60 (the median divides the area of the triangle into two equal parts)
+
<math>A[\Delta ABE] = A[\Delta AED] = 60</math> (the median divides the area of the triangle into two equal parts)
  
  
Construction: Draw a circumcircle around ∆ABD with BD as is diameter. Extend AF to G such that it meets the circle at G. Draw line BG.
+
Construction: Draw a circumcircle around <math>\Delta ABD</math> with <math>BD</math> as is diameter. Extend <math>AF</math> to <math>G</math> such that it meets the circle at <math>G</math>. Draw line <math>BG</math>.
  
  
Implies <math>A[∆ABD] = A[∆ABG] = 120</math> (Since [∎ABGD] is cyclic)
+
<math>A[\Delta ABD] = A[\Delta ABG] = 120</math> (Since <math>\square ABGD</math> is cyclic)
  
  
But A[∆ABE] is common in both with an area of 60.
+
But <math>A[\Delta ABE]</math> is common in both with an area of 60. So, <math>A[\Delta AED] = A[\Delta BEG]</math>.
  
So A[∆AED] = A[∆BEG]
+
\therefore <math>A[\Delta AED] \cong A[\Delta BEG]</math> (SAS Congruency Theorem).
  
Therefore A[∆AED] ≅ A[∆BEG] (SAS)
+
In <math>\Delta AED</math>, let <math>DI</math> be the median of <math>\Delta AED</math>.
  
In ∆AED, let DI be the median of ∆AED.
+
Which means <math>A[\Delta AID] = 30 = A[\Delta EID]</math>
  
Which means A[∆AID] = 30 = A[∆EID]
 
  
 +
Rotate <math>\Delta DEA</math> to meet <math>D</math> at <math>B</math> and <math>A</math> at <math>G</math>. <math>DE</math> will fit exactly in <math>BE</math> (both are radii of the circle). From the above solutions, <math>\frac{AE}{EF} = 2:1</math>.
  
In ∆BEG, we need to find if BF is the median of ∆BEG.
+
<math>AE</math> is a radius and <math>EF</math> is half of it implies <math>EF</math> = <math>\frac{radius}{2}</math>.
  
Therefore, rotate ∆DEA to meet D at B and A at F.
+
Which means <math>A[\Delta BEF] \cong A[\Delta DEI]</math>
  
 +
Thus <math>A[\Delta BEF] = 30</math>
  
We know that DE will fit exactly in BE (DE and BE are the radii of the circle).
 
  
And due to the above solutions we know <math>\frac{AE}{EF} = 2:1</math>.
+
~phoenixfire & flamewavelight
 
 
<math>AE</math> is a radius and if <math>EF</math> is half of it implies <math>EF</math> = <math>\frac{radius}{2}</math>.
 
 
 
Which means A[∆BEF] ≅ A[∆DEI]
 
 
 
Thus A[∆BEF]= <math>30</math>.
 
  
 
==See Also==
 
==See Also==

Revision as of 13:26, 21 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 $X$ on $\overline{AF}$ such that $XD$ is parallel to $BC$. That makes triangles $BEF$ and $EXD$ congruent since $BE = ED$. $FC=3XD$ so $BC=4BF$. Since $AF=3EF$ ($XE=EF$ and $AX=\frac13 AF$, so $XE=EF=\frac13 AF$), the altitude of triangle $BEF$ is equal to $\frac{1}{3}$ of the altitude of $ABC$. The area of $ABC$ is $360$, so the area of $BEF=\frac{1}{3} \cdot \frac{1}{4} \cdot 360=\boxed{(B) 30}$ ~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

Solution 3

$\frac{BF}{FC}$ is equal to $\frac{\textrm{The area of triangle ABE}}{\textrm{The area of triangle ACE}}$. The area of triangle $ABE$ is equal to $60$ because it is equal to on half of the area of triangle $ABD$, which is equal to one third of the area of triangle $ABC$, which is $360$. The area of triangle $ACE$ is the sum of the areas of triangles $AED$ and $CED$, which is respectively $60$ and $120$. So, $\frac{BF}{FC}$ is equal to $\frac{60}{180}$=$\frac{1}{3}$, so the area of triangle $ABF$ is $90$. That minus the area of triangle $ABE$ is $\boxed{(B) 30}$. ~~SmileKat32

Solution 4 (Similar Triangles)

Extend $\overline{BD}$ to $G$ such that $\overline{AG} \parallel \overline{BC}$ as shown: [asy] size(8cm); pair A, B, C, D, E, F, G; B = (0,0); A = (2, 3); C = (5, 0); D = (3, 2); E = (1.5, 1); F = (1.25, 0); G = (4.5, 3);  draw(A--B--C--A--G--B); draw(A--F); label("$A$", A, N); label("$B$", B, WSW); label("$C$", C, ESE); label("$D$", D, dir(0)*1.5); label("$E$", E, SE); label("$F$", F, S); label("$G$", G, ENE); [/asy] Then $\triangle ADG \sim \triangle CDB$ and $\triangle AEG \sim \triangle FEB$. Since $CD = 2AD$, triangle $CDB$ has four times the area of triangle $ADG$. Since $[CDB] = 240$, we get $[ADG] = 60$.

Since $[AED]$ is also $60$, we have $ED = DG$ because triangles $AED$ and $ADG$ have the same height and same areas and so their bases must be the congruent. Thus triangle $AEG$ has twice the side lengths and therefore four times the area of triangle $BEF$, giving $[BEF] = (60+60)/4 = \boxed{\textbf{(B) }30}$.

[asy] size(8cm); pair A, B, C, D, E, F, G; B = (0,0); A = (2, 3); C = (5, 0); D = (3, 2); E = (1.5, 1); F = (1.25, 0); G = (4.5, 3);  draw(A--B--C--A--G--B); draw(A--F); label("$A$", A, N); label("$B$", B, WSW); label("$C$", C, ESE); label("$D$", D, dir(0)*1.5); label("$E$", E, SE); label("$F$", F, S); label("$G$", G, ENE); label("$60$", (A+E+D)/3); label("$60$", (A+E+B)/3); label("$60$", (A+G+D)/3); label("$30$", (B+E+F)/3); [/asy] (Credit to MP8148 for the idea)

Solution 5 (Area Ratios)

[asy] size(8cm); pair A, B, C, D, E, F; B = (0,0); A = (2, 3); C = (5, 0); D = (3, 2); E = (1.5, 1); F = (1.25, 0);  draw(A--B--C--A--D--B); draw(A--F); draw(E--C); label("$A$", A, N); label("$B$", B, WSW); label("$C$", C, ESE); label("$D$", D, dir(0)*1.5); label("$E$", E, SSE); label("$F$", F, S); label("$60$", (A+E+D)/3); label("$60$", (A+E+B)/3); label("$120$", (D+E+C)/3); label("$x$", (B+E+F)/3); label("$120-x$", (F+E+C)/3); [/asy] As before we figure out the areas labeled in the diagram. Then we note that \[\dfrac{EF}{AE} = \dfrac{x}{60} = \dfrac{120-x}{180}.\]Solving gives $x = \boxed{\textbf{(B) }30}$. (Credit to scrabbler94 for the idea)

Solution 6(Coordbash)

Let $ADB$ be a right triangle, and $BD=CD$

Let $A=(-2\sqrt{30}, 0)$

$B=(0, 4\sqrt{30})$

$C=(4\sqrt{30}, 0)$

$D=(0, 0)$

$E=(0, 2\sqrt{30})$

The line $\overleftrightarrow{AE}$ can be described with the equation $y=x-2\sqrt{30}$

The line $\overleftrightarrow{BC}$ can be described with $x+y=4\sqrt{30}$

Solving, we get $x=3\sqrt{30}$ and $y=\sqrt{30}$

Now we can find $EF=BF=2\sqrt{15}$

$[\bigtriangleup EBF]=\frac{(2\sqrt{15})^2}{2}=\boxed{(B) 30}\blacksquare$

-Trex4days

Solution 7

$A[\Delta ABD]: A[\Delta DBC] :: 1:2 :: 120:240$


$A[\Delta ABE] = A[\Delta AED] = 60$ (the median divides the area of the triangle into two equal parts)


Construction: Draw a circumcircle around $\Delta ABD$ with $BD$ as is diameter. Extend $AF$ to $G$ such that it meets the circle at $G$. Draw line $BG$.


$A[\Delta ABD] = A[\Delta ABG] = 120$ (Since $\square ABGD$ is cyclic)


But $A[\Delta ABE]$ is common in both with an area of 60. So, $A[\Delta AED] = A[\Delta BEG]$.

\therefore $A[\Delta AED] \cong A[\Delta BEG]$ (SAS Congruency Theorem).

In $\Delta AED$, let $DI$ be the median of $\Delta AED$.

Which means $A[\Delta AID] = 30 = A[\Delta EID]$


Rotate $\Delta DEA$ to meet $D$ at $B$ and $A$ at $G$. $DE$ will fit exactly in $BE$ (both are radii of the circle). From the above solutions, $\frac{AE}{EF} = 2:1$.

$AE$ is a radius and $EF$ is half of it implies $EF$ = $\frac{radius}{2}$.

Which means $A[\Delta BEF] \cong A[\Delta DEI]$

Thus $A[\Delta BEF] = 30$


~phoenixfire & flamewavelight

See Also

2019 AMC 8 (ProblemsAnswer KeyResources)
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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