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Difference between revisions of "2001 AMC 12 Problems/Problem 22"

m (→‎Solution 5)
m (→‎Solution 5)
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<math>[CEF] = \frac{[ABCD]}{4} = \frac{35}{2}</math>
 
<math>[CEF] = \frac{[ABCD]}{4} = \frac{35}{2}</math>
  
−
Since <math>\triangle CEH \sim \triangle AFH</math>, <math>\frac{HE}{HF} = \frac{CE}{AF} = \frac{3}{2}</math>, <math>\frac{HE}{EF} = \frac{3}{5}</math>
+
<math>\triangle CEH \sim \triangle AFH</math>, <math>\frac{HE}{HF} = \frac{CE}{AF} = \frac{3}{2}</math>, <math>\frac{HE}{EF} = \frac{3}{5}</math>
  
−
<math>[CEH] = \frac{HE}{HF} \cdot [FEC] = \frac{3}{5} \cdot \frac{35}{2} = \frac{21}{2}</math>
+
<math>[CEH] = \frac{HE}{EF} \cdot [CEF] = \frac{3}{5} \cdot \frac{35}{2} = \frac{21}{2}</math>
  
−
Since <math>\triangle CEH \sim \triangle AFH</math>, <math>\frac{AH}{HC} = \frac{AF}{CE} = \frac{2}{3}</math>, <math>\frac{AH}{AC} = \frac{2}{5}</math>, <math>\frac{CH}{AC} = \frac{3}{5}</math>
+
<math>\triangle CEH \sim \triangle AFH</math>, <math>\frac{AH}{HC} = \frac{AF}{CE} = \frac{2}{3}</math>, <math>\frac{AH}{AC} = \frac{2}{5}</math>, <math>\frac{CH}{AC} = \frac{3}{5}</math>
  
−
Since <math>\triangle CEJ \sim \triangle AGJ</math>, <math>\frac{AJ}{JC} = \frac{AG}{CE} = \frac{4}{3}</math>, <math>\frac{AJ}{AC} = \frac{4}{7}</math>
+
<math>\triangle CEJ \sim \triangle AGJ</math>, <math>\frac{AJ}{JC} = \frac{AG}{CE} = \frac{4}{3}</math>, <math>\frac{AJ}{AC} = \frac{4}{7}</math>
  
 
<math>\frac{HJ}{AC} = \frac{AJ}{AC} - \frac{AH}{AC} = \frac{4}{7} - \frac{2}{5} = \frac{6}{35}</math>
 
<math>\frac{HJ}{AC} = \frac{AJ}{AC} - \frac{AH}{AC} = \frac{4}{7} - \frac{2}{5} = \frac{6}{35}</math>
  
−
<math>\frac{HJ}{AC} = \frac{HJ}{AC} \cdot \frac{AC}{CH} = \frac{6}{35} \cdot \frac{5}{3} = \frac{2}{7}</math>
+
<math>\frac{HJ}{CH} = \frac{HJ}{AC} \cdot \frac{AC}{CH} = \frac{6}{35} \cdot \frac{5}{3} = \frac{2}{7}</math>
  
−
Because <math>\triangle EHJ</math> and <math>\triangle CEH</math> share the same height to side <math>CH</math>, <math>[EHJ] = [CEH] \cdot \frac{HJ}{AC} = \frac{21}{2} \cdot \frac{2}{7} = \boxed{\textbf{(C) }3}</math>
+
<math>[EHJ] = \frac{HJ}{CH} \cdot [CEH] = \frac{2}{7} \cdot \frac{21}{2} = \boxed{\textbf{(C) }3}</math>
  
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]

Revision as of 09:38, 25 August 2022

Problem

In rectangle $ABCD$, points $F$ and $G$ lie on $AB$ so that $AF=FG=GB$ and $E$ is the midpoint of $\overline{DC}$. Also, $\overline{AC}$ intersects $\overline{EF}$ at $H$ and $\overline{EG}$ at $J$. The area of the rectangle $ABCD$ is $70$. Find the area of triangle $EHJ$.

$\text{(A) }\frac {5}{2} \qquad \text{(B) }\frac {35}{12} \qquad \text{(C) }3 \qquad \text{(D) }\frac {7}{2} \qquad \text{(E) }\frac {35}{8}$

[asy] unitsize(0.5cm); defaultpen(0.8); pair A=(0,0), B=(10,0), C=(10,7), D=(0,7), E=(C+D)/2, F=(2*A+B)/3, G=(A+2*B)/3; pair H = intersectionpoint(A--C,E--F); pair J = intersectionpoint(A--C,E--G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,S); label("$G$",G,S); label("$H$",H,SE); label("$J$",J,ESE); filldraw(E--H--J--cycle,lightgray,black); draw(H--D, dashed); [/asy]


Solution 1

Note that the triangles $AFH$ and $CEH$ are similar, as they have the same angles. Hence $\frac {AH}{HC} = \frac{AF}{EC} = \frac 23$.

Also, triangles $AGJ$ and $CEJ$ are similar, hence $\frac {AJ}{JC} = \frac {AG}{EC} = \frac 43$.

We can now compute $[EHJ]$ as $[ACD]-[AHD]-[DEH]-[EJC]$. We have:

  • $[ACD]=\frac{[ABCD]}2 = 35$.
  • $[AHD]$ is $2/5$ of $[ACD]$, as these two triangles have the same base $AD$, and $AH$ is $2/5$ of $AC$, therefore also the height from $H$ onto $AD$ is $2/5$ of the height from $C$. Hence $[AHD]=14$.
  • $[HED]$ is $3/10$ of $[ACD]$, as the base $ED$ is $1/2$ of the base $CD$, and the height from $H$ is $3/5$ of the height from $A$. Hence $[HED]=\frac {21}2$.
  • $[JEC]$ is $3/14$ of $[ACD]$ for similar reasons, hence $[JEC]=\frac{15}2$.

Therefore $[EHJ]=[ACD]-[AHD]-[DEH]-[EJC]=35-14-\frac {21}2-\frac{15}2 = \boxed{3}$.

Solution 2

As in the previous solution, we note the similar triangles and prove that $H$ is in $2/5$ and $J$ in $4/7$ of $AC$.

We can then compute that $HJ = AC \cdot \left( \frac 47 - \frac 25 \right) = AC \cdot \frac{6}{35}$.

As $E$ is the midpoint of $CD$, the height from $E$ onto $AC$ is $1/2$ of the height from $D$ onto $AC$. Therefore we have $[EHJ] = \frac{6}{35} \cdot \frac 12 \cdot [ACD] = \frac 3{35} \cdot 35 = \boxed{3}$.

Solution 3

Because we see that there are only lines and there is a rectangle, we can coordbash (place this figure on coordinates). Because this is a general figure, we can assume the sides are $7$ and $10$ (or any other two positive real numbers that multiply to 70). We can find $H$ and $J$ by intersecting lines, and then we calculate the area of $EHJ$ using shoelace formula. This yields $\boxed{3}$.

Solution 4

Note that triangle $AFH$ is similar to triangle $CEH$ with ratio $\frac{2}{3}$. Similarly, triangle $AGJ$ is similar to triangle $ECJ$ with ratio $\frac{4}{3}$. Thus, if $AC = a$ then we know that $AH = \frac{2}{5}a$ and $JC = \frac{3}{7}a$ meaning $HJ = \frac{6}{35}a$ and thus the ratio of $HJ$ to $JC$ is $\frac{\frac{6}{35}}{\frac{3}{7}} = \frac{2}{5}$ which equals the ratio of the areas of $HJE$ to $JEC$. If $y = AD, x = DC$, then we know that $JEC = \text{(altitude from J to EC)} \cdot EC = \frac{3}{7}y \cdot \frac{1}{2}x \cdot \frac{1}{2}$ and since $xy = 70$ and we want to find $\frac{2}{5}$ of this, we get our answer is $\frac{2}{5} \cdot \frac{3}{7} \cdot \frac{1}{2} \cdot 70 \cdot \frac{1}{2} = \boxed{3}$. -SuperJJ

Solution 5

$[CEF] = \frac{[ABCD]}{4} = \frac{35}{2}$

$\triangle CEH \sim \triangle AFH$, $\frac{HE}{HF} = \frac{CE}{AF} = \frac{3}{2}$, $\frac{HE}{EF} = \frac{3}{5}$

$[CEH] = \frac{HE}{EF} \cdot [CEF] = \frac{3}{5} \cdot \frac{35}{2} = \frac{21}{2}$

$\triangle CEH \sim \triangle AFH$, $\frac{AH}{HC} = \frac{AF}{CE} = \frac{2}{3}$, $\frac{AH}{AC} = \frac{2}{5}$, $\frac{CH}{AC} = \frac{3}{5}$

$\triangle CEJ \sim \triangle AGJ$, $\frac{AJ}{JC} = \frac{AG}{CE} = \frac{4}{3}$, $\frac{AJ}{AC} = \frac{4}{7}$

$\frac{HJ}{AC} = \frac{AJ}{AC} - \frac{AH}{AC} = \frac{4}{7} - \frac{2}{5} = \frac{6}{35}$

$\frac{HJ}{CH} = \frac{HJ}{AC} \cdot \frac{AC}{CH} = \frac{6}{35} \cdot \frac{5}{3} = \frac{2}{7}$

$[EHJ] = \frac{HJ}{CH} \cdot [CEH] = \frac{2}{7} \cdot \frac{21}{2} = \boxed{\textbf{(C) }3}$

~isabelchen

See Also

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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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