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Difference between revisions of "2018 AMC 10A Problems/Problem 24"

m (→‎Solution 7: Barycentrics)
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The area of quadrilateral <math>FDBG</math> is the area of <math>\bigtriangleup ABG</math> minus the area of <math>\bigtriangleup ADF</math>. Notice, <math>\overline{DE} || \overline{BC}</math>, so <math>\bigtriangleup ABG \sim \bigtriangleup ADF</math>, and since <math>\overline{AD}:\overline{AB}=1:2</math>, the area of <math>\bigtriangleup ADF:\bigtriangleup ABG=(1:2)^2=1:4</math>. Given that the area of <math>\bigtriangleup ABC</math> is <math>120</math>, using <math>\frac{bh}{2}</math> on side <math>AB</math> yields <math>\frac{50h}{2}=120\implies h=\frac{240}{50}=\frac{24}{5}</math>. Using the Angle Bisector Theorem, <math>\overline{BG}:\overline{BC}=50:(10+50)=5:6</math>, so the height of <math>\bigtriangleup ABG: \bigtriangleup ACB=5:6</math>. Therefore our answer is <math>\big[ FDBG\big] = \big[ABG\big]-\big[ ADF\big] = \big[ ABG\big]\big(1-\frac{1}{4}\big)=\frac{3}{4}\cdot \frac{bh}{2}=\frac{3}{8}\cdot 50\cdot \frac{5}{6}\cdot \frac{24}{5}=\frac{3}{8}\cdot 200=\boxed{75}</math>
 
The area of quadrilateral <math>FDBG</math> is the area of <math>\bigtriangleup ABG</math> minus the area of <math>\bigtriangleup ADF</math>. Notice, <math>\overline{DE} || \overline{BC}</math>, so <math>\bigtriangleup ABG \sim \bigtriangleup ADF</math>, and since <math>\overline{AD}:\overline{AB}=1:2</math>, the area of <math>\bigtriangleup ADF:\bigtriangleup ABG=(1:2)^2=1:4</math>. Given that the area of <math>\bigtriangleup ABC</math> is <math>120</math>, using <math>\frac{bh}{2}</math> on side <math>AB</math> yields <math>\frac{50h}{2}=120\implies h=\frac{240}{50}=\frac{24}{5}</math>. Using the Angle Bisector Theorem, <math>\overline{BG}:\overline{BC}=50:(10+50)=5:6</math>, so the height of <math>\bigtriangleup ABG: \bigtriangleup ACB=5:6</math>. Therefore our answer is <math>\big[ FDBG\big] = \big[ABG\big]-\big[ ADF\big] = \big[ ABG\big]\big(1-\frac{1}{4}\big)=\frac{3}{4}\cdot \frac{bh}{2}=\frac{3}{8}\cdot 50\cdot \frac{5}{6}\cdot \frac{24}{5}=\frac{3}{8}\cdot 200=\boxed{75}</math>
  
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==Solution 5: Trig ==
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==Solution 5 (Trigonometry) ==
 
We try to find the area of quadrilateral <math>FDBG</math> by subtracting the area outside the quadrilateral but inside triangle <math>ABC</math>. Note that the area of <math>\triangle ADE</math> is equal  to <math>\frac{1}{2} \cdot 25 \cdot 5 \cdot \sin{A}</math> and the area of triangle <math>ABC</math> is equal to <math>\frac{1}{2} \cdot 50 \cdot 10 \cdot \sin A</math>. The ratio <math>\frac{[ADE]}{[ABC]}</math> is thus equal to <math>\frac{1}{4}</math> and the area of triangle <math>ADE</math> is <math>\frac{1}{4} \cdot 120 = 30</math>. Let side <math>BC</math> be equal to <math>6x</math>, then <math>BG = 5x, GC = x</math> by the angle bisector theorem. Similarly, we find the area of triangle <math>AGC</math> to be <math>\frac{1}{2} \cdot 10 \cdot x \cdot \sin C</math> and the area of triangle <math>ABC</math> to be <math>\frac{1}{2} \cdot 6x \cdot 10 \cdot \sin C</math>. A ratio between these two triangles yields <math>\frac{[ACG]}{[ABC]} = \frac{x}{6x} = \frac{1}{6}</math>, so <math>[AGC] = 20</math>. Now we just need to find the area of triangle <math>AFE</math> and subtract it from the combined areas of <math>[ADE]</math> and <math>[ACG]</math>, since we count it twice. Note that the angle bisector theorem also applies for <math>\triangle ADE</math> and <math>\frac{AE}{AD} = \frac{1}{5}</math>, so thus <math>\frac{EF}{ED} = \frac{1}{6}</math> and we find <math>[AFE] = \frac{1}{6} \cdot 30 = 5</math>, and the area outside <math>FDBG</math> must be <math> [ADE] + [AGC] - [AFE] = 30 + 20 - 5 = 45</math>, and we finally find  <math>[FDBG] = [ABC] - 45 = 120 -45 = \boxed{75}</math>, and we are done.  
 
We try to find the area of quadrilateral <math>FDBG</math> by subtracting the area outside the quadrilateral but inside triangle <math>ABC</math>. Note that the area of <math>\triangle ADE</math> is equal  to <math>\frac{1}{2} \cdot 25 \cdot 5 \cdot \sin{A}</math> and the area of triangle <math>ABC</math> is equal to <math>\frac{1}{2} \cdot 50 \cdot 10 \cdot \sin A</math>. The ratio <math>\frac{[ADE]}{[ABC]}</math> is thus equal to <math>\frac{1}{4}</math> and the area of triangle <math>ADE</math> is <math>\frac{1}{4} \cdot 120 = 30</math>. Let side <math>BC</math> be equal to <math>6x</math>, then <math>BG = 5x, GC = x</math> by the angle bisector theorem. Similarly, we find the area of triangle <math>AGC</math> to be <math>\frac{1}{2} \cdot 10 \cdot x \cdot \sin C</math> and the area of triangle <math>ABC</math> to be <math>\frac{1}{2} \cdot 6x \cdot 10 \cdot \sin C</math>. A ratio between these two triangles yields <math>\frac{[ACG]}{[ABC]} = \frac{x}{6x} = \frac{1}{6}</math>, so <math>[AGC] = 20</math>. Now we just need to find the area of triangle <math>AFE</math> and subtract it from the combined areas of <math>[ADE]</math> and <math>[ACG]</math>, since we count it twice. Note that the angle bisector theorem also applies for <math>\triangle ADE</math> and <math>\frac{AE}{AD} = \frac{1}{5}</math>, so thus <math>\frac{EF}{ED} = \frac{1}{6}</math> and we find <math>[AFE] = \frac{1}{6} \cdot 30 = 5</math>, and the area outside <math>FDBG</math> must be <math> [ADE] + [AGC] - [AFE] = 30 + 20 - 5 = 45</math>, and we finally find  <math>[FDBG] = [ABC] - 45 = 120 -45 = \boxed{75}</math>, and we are done.  
  
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==Solution 6: Areas ==
+
==Solution 6 (Areas) ==
 
<asy>
 
<asy>
 
draw((0,0)--(1,3)--(5,0)--cycle);
 
draw((0,0)--(1,3)--(5,0)--cycle);
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Additionally, since all points on <math>\overline{AG}</math> are characterized by <math>(t:1:5)</math>, we may plug in for <math>x,y,z</math> to get <math>t=6</math>. Thus, we have <math>F=(6:1:5).</math> Now, we homogenize the coordinates for <math>F D, B, G</math> to get <math>F=(\frac{1}{2}, \frac{5}{12}, \frac{1}{12})</math>, <math>D=(\frac{1}{2}, \frac{1}{2}, 0)</math>, <math>B=(0,1,0)</math>, <math>G=(0, \frac{1}{6}, \frac{5}{6})</math>
 
Additionally, since all points on <math>\overline{AG}</math> are characterized by <math>(t:1:5)</math>, we may plug in for <math>x,y,z</math> to get <math>t=6</math>. Thus, we have <math>F=(6:1:5).</math> Now, we homogenize the coordinates for <math>F D, B, G</math> to get <math>F=(\frac{1}{2}, \frac{5}{12}, \frac{1}{12})</math>, <math>D=(\frac{1}{2}, \frac{1}{2}, 0)</math>, <math>B=(0,1,0)</math>, <math>G=(0, \frac{1}{6}, \frac{5}{6})</math>
−
 
−
 
  
 
Splitting <math>[FBGD]</math> into <math>[ DBG ] + [ FDG],</math> we may now evaluate the two determinants:
 
Splitting <math>[FBGD]</math> into <math>[ DBG ] + [ FDG],</math> we may now evaluate the two determinants:

Revision as of 09:59, 17 June 2023

The following problem is from both the 2018 AMC 10A #24 and 2018 AMC 12A #18, so both problems redirect to this page.

Problem

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?

$\textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad$

Solution 1

Let $BC = a$, $BG = x$, $GC = y$, and the length of the perpendicular to $BC$ through $A$ be $h$. By angle bisector theorem, we have that \[\frac{50}{x} = \frac{10}{y},\] where $y = -x+a$. Therefore substituting we have that $BG=\frac{5a}{6}$. By similar triangles, we have that $DF=\frac{5a}{12}$, and the height of this trapezoid is $\frac{h}{2}$. Then, we have that $\frac{ah}{2}=120$. We wish to compute $\frac{5a}{8}\cdot\frac{h}{2}$, and we have that it is $\boxed{75}$ by substituting.

Solution 2

For this problem, we have $\triangle{ADE}\sim\triangle{ABC}$ because of SAS and $DE = \frac{BC}{2}$. Therefore, $\bigtriangleup ADE$ is a quarter of the area of $\bigtriangleup ABC$, which is $30$. Subsequently, we can compute the area of quadrilateral $BDEC$ to be $120 - 30 = 90$. Using the angle bisector theorem in the same fashion as the previous problem, we get that $\overline{BG}$ is $5$ times the length of $\overline{GC}$. We want the larger piece, as described by the problem. Because the heights are identical, one area is $5$ times the other, and $\frac{5}{6} \cdot 90 = \boxed{75}$.

Solution 3

The ratio of the $\overline{BG}$ to $\overline{GC}$ is $5:1$ by the Angle Bisector Theorem, so area of $\bigtriangleup ABG$ to the area of $\bigtriangleup ACG$ is also $5:1$ (They have the same height). Therefore, the area of $\bigtriangleup ABG$ is $\frac{5}{5+1}\times120=100$. Since $\overline{DE}$ is the midsegment of $\bigtriangleup ABC$, so $\overline{DF}$ is the midsegment of $\bigtriangleup ABG$ . Thus, the ratio of the area of $\bigtriangleup ADF$ to the area of $\bigtriangleup ABG$ is $1:4$, so the area of $\bigtriangleup ACG$ is $\frac{1}{4}\times100=25$. Therefore, the area of quadrilateral $FDBG$ is $[ABG]-[ADF]=100-25=\boxed{75}$

Solution 4

The area of quadrilateral $FDBG$ is the area of $\bigtriangleup ABG$ minus the area of $\bigtriangleup ADF$. Notice, $\overline{DE} || \overline{BC}$, so $\bigtriangleup ABG \sim \bigtriangleup ADF$, and since $\overline{AD}:\overline{AB}=1:2$, the area of $\bigtriangleup ADF:\bigtriangleup ABG=(1:2)^2=1:4$. Given that the area of $\bigtriangleup ABC$ is $120$, using $\frac{bh}{2}$ on side $AB$ yields $\frac{50h}{2}=120\implies h=\frac{240}{50}=\frac{24}{5}$. Using the Angle Bisector Theorem, $\overline{BG}:\overline{BC}=50:(10+50)=5:6$, so the height of $\bigtriangleup ABG: \bigtriangleup ACB=5:6$. Therefore our answer is $\big[ FDBG\big] = \big[ABG\big]-\big[ ADF\big] = \big[ ABG\big]\big(1-\frac{1}{4}\big)=\frac{3}{4}\cdot \frac{bh}{2}=\frac{3}{8}\cdot 50\cdot \frac{5}{6}\cdot \frac{24}{5}=\frac{3}{8}\cdot 200=\boxed{75}$

Solution 5 (Trigonometry)

We try to find the area of quadrilateral $FDBG$ by subtracting the area outside the quadrilateral but inside triangle $ABC$. Note that the area of $\triangle ADE$ is equal to $\frac{1}{2} \cdot 25 \cdot 5 \cdot \sin{A}$ and the area of triangle $ABC$ is equal to $\frac{1}{2} \cdot 50 \cdot 10 \cdot \sin A$. The ratio $\frac{[ADE]}{[ABC]}$ is thus equal to $\frac{1}{4}$ and the area of triangle $ADE$ is $\frac{1}{4} \cdot 120 = 30$. Let side $BC$ be equal to $6x$, then $BG = 5x, GC = x$ by the angle bisector theorem. Similarly, we find the area of triangle $AGC$ to be $\frac{1}{2} \cdot 10 \cdot x \cdot \sin C$ and the area of triangle $ABC$ to be $\frac{1}{2} \cdot 6x \cdot 10 \cdot \sin C$. A ratio between these two triangles yields $\frac{[ACG]}{[ABC]} = \frac{x}{6x} = \frac{1}{6}$, so $[AGC] = 20$. Now we just need to find the area of triangle $AFE$ and subtract it from the combined areas of $[ADE]$ and $[ACG]$, since we count it twice. Note that the angle bisector theorem also applies for $\triangle ADE$ and $\frac{AE}{AD} = \frac{1}{5}$, so thus $\frac{EF}{ED} = \frac{1}{6}$ and we find $[AFE] = \frac{1}{6} \cdot 30 = 5$, and the area outside $FDBG$ must be $[ADE] + [AGC] - [AFE] = 30 + 20 - 5 = 45$, and we finally find $[FDBG] = [ABC] - 45 = 120 -45 = \boxed{75}$, and we are done.

Solution 6 (Areas)

[asy] draw((0,0)--(1,3)--(5,0)--cycle); draw((0,0)--(2,2.25)); draw((0.5,1.5)--(2.5,0)); label("A",(0,0),SW); label("B",(5,0),SE); label("C",(1,3),N); label("G",(2,2.25),NE); label("D",(2.5,0),S); label("E",(0.5,1.5),NW); label("3Y",(2.5,0.75),N); label("Y",(1,0.2),N); label("X",(0.5,0.5),N); label("3X",(1.25,1.75),N); [/asy] Give triangle $AEF$ area X. Then, by similarity, since $\frac{AC}{AE} = \frac{2}{1}$, $ACG$ has area 4X. Thus, $FGCE$ has area 3X. Doing the same for triangle $AGB$, we get that triangle $AFD$ has area Y and quadrilateral $GFDB$ has area 3Y. Since $AEF$ has the same height as $AFD$, the ratios of the areas is equal to the ratios of the bases. Because of the Angle Bisector Theorem, $\frac{CG}{GB} = \frac{1}{5}$. So, $\frac{[AEF]}{[AFD]} = \frac{1}{5}$. Since $AEF$ has area X, we can write the equation 5X = Y and substitute 5X for Y. [asy] draw((0,0)--(1,3)--(5,0)--cycle); draw((0,0)--(2,2.25)); draw((0.5,1.5)--(2.5,0)); label("A",(0,0),SW); label("B",(5,0),SE); label("C",(1,3),N); label("G",(2,2.25),NE); label("D",(2.5,0),S); label("E",(0.5,1.5),NW); label("",(2.5,0.75),N); label("",(1,0.2),N); label("F", (1, 1.5), N); label("",(0.5,1.5),N); label("",(1.25,1.75),N); [/asy] Now we can solve for X by adding up all the sums. X + 3X + 5X + 15X = 120, so X = 5. Since we want to find $GFDB$, we substitute 5 for 15X to get $\boxed{75}$. $\sim$krishkhushi09

Solution 7 (Barycentrics)

Let our reference triangle be $\triangle ABC$. Consequently, we have $A=(1,0,0)$, $B=(0,1,0)$, $C=(0,0,1).$ Since $D$ is the midpoint of $\overline{AB}$, we have that $D=(1:1:0)$. Similarly, we have $E=(1:0:1).$ Hence, the line through $D$ and $E$ is given by the equation

\[0 = \begin{vmatrix} x & y & z\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{vmatrix}\]

Additionally, since all points on $\overline{AG}$ are characterized by $(t:1:5)$, we may plug in for $x,y,z$ to get $t=6$. Thus, we have $F=(6:1:5).$ Now, we homogenize the coordinates for $F D, B, G$ to get $F=(\frac{1}{2}, \frac{5}{12}, \frac{1}{12})$, $D=(\frac{1}{2}, \frac{1}{2}, 0)$, $B=(0,1,0)$, $G=(0, \frac{1}{6}, \frac{5}{6})$

Splitting $[FBGD]$ into $[ DBG ] + [ FDG],$ we may now evaluate the two determinants:

\[\begin{vmatrix} \frac{1}{2} & \frac{1}{2} & 0\\ 0 & 1 & 0\\ 0 & \frac{1}{6} & \frac{5}{6} \end{vmatrix}\] \[\begin{vmatrix} \frac{1}{2} & \frac{1}{12} & \frac{5}{12}\\ \frac{1}{2} & \frac{1}{2} & 0\\ 0 & \frac{5}{6} & \frac{1}{6} \end{vmatrix}.\]

After simplification, we get $\frac{5}{12}$ and $\frac{5}{24}$, respectively. Summing, we get $\frac{15}{24}.$ Hence, $[FBGD]=\frac{15}{24} \cdot 120 = \fbox{75}.$ $\sim$Math0323

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2018amc10a/469

~ dolphin7

Video Solution by OmegaLearn

https://youtu.be/4_x1sgcQCp4?t=4898

~ pi_is_3.14

See Also

2018 AMC 10A (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 10 Problems and Solutions
2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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

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