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

Latest revision as of 01:27, 29 August 2023

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 $EHJ$ .

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

Solution 1

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

Solution 6 (Mass Points)

We need one more pair of ratios to fully define our mass point system. Let's use $\triangle AFH \sim \triangle CEH\implies EH:HF = 3:2$ and now do mass points on $\triangle AEG$ :

$[asy] size(250); pair A = (0,0), B = (10,0), C = (10,7), D = (0,7); pair F = (10/3,0), G = (20/3,0), E = (5,7); pair H = intersectionpoint(A--C, E--F); pair J = intersectionpoint(A--C, E--G); filldraw(A--E--G--cycle, rgb(1,1,1)+opacity(0.3), red+2bp); draw(A--B--C--D--cycle); draw(A--C); draw(E--F); draw(E--G); draw(A--E, dashed); draw(E--B, dashed); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, N); dot("$F$", F, S); dot("$G$", G, S); dot("$H$", H, ESE); dot("$J$", J, W); // mass point labels pair mass = A + SW; label(scale(0.8)*"$3$", mass, UnFill); draw(circle(mass, .4), linewidth(1)); pair mass = F + S; label(scale(0.8)*"$6$", mass, UnFill); draw(circle(mass, .4), linewidth(1)); pair mass = G + SE; label(scale(0.8)*"$3$", mass, UnFill); draw(circle(mass, .4), linewidth(1)); pair mass = J + .7*ESE; label(scale(0.8)*"$7$", mass, UnFill); draw(circle(mass, .4), linewidth(1)); pair mass = E + N; label(scale(0.8)*"$4$", mass, UnFill); draw(circle(mass, .4), linewidth(1)); pair mass = H + .7*WNW; label(scale(0.8)*"$10$", mass, UnFill); draw(circle(mass, .4), linewidth(1)); [/asy]$

Now it's just a standard "Area Reduction by Ratios"™ problem going from:

$[ABCD]\xrightarrow[]{\frac{1}{2}}[AEB]\xrightarrow[]{\frac{2}{3}}[AEG]\xrightarrow[]{\frac{3}{7}}[AEJ]\xrightarrow[]{\frac{3}{10}}[HEJ]$

or,

$70 \cdot \frac{1}{2}\cdot \frac{2}{3}\cdot \frac{3}{7}\cdot \frac{3}{10} = \boxed{\textbf{(C) }3}$

~ proloto

@@ Line 15: / Line 15: @@
 </math>
-[[Category: Introductory Geometry Problems]]
+== Solution 1 ==
 <asy>
 unitsize(0.5cm);
@@ Line 38: / Line 37: @@
 draw(H--D, dashed);
 </asy>
-== Solution 1 ==
 Note that the triangles <math>AFH</math> and <math>CEH</math> are similar, as they have the same angles. Hence <math>\frac {AH}{HC} = \frac{AF}{EC} = \frac 23</math>.
@@ Line 69: / Line 64: @@
 Note that triangle <math>AFH</math> is similar to triangle <math>CEH</math> with ratio <math>\frac{2}{3}</math>. Similarly, triangle <math>AGJ</math> is similar to triangle <math>ECJ</math> with ratio <math>\frac{4}{3}</math>. Thus, if <math>AC = a</math> then we know that <math>AH = \frac{2}{5}a</math> and <math>JC = \frac{3}{7}a</math> meaning <math>HJ = \frac{6}{35}a</math> and thus the ratio of <math>HJ</math> to <math>JC</math> is <math>\frac{\frac{6}{35}}{\frac{3}{7}} = \frac{2}{5}</math> which equals the ratio of the areas of <math>HJE</math> to <math>JEC</math>. If <math>y = AD, x = DC</math>, then we know that <math>JEC = \text{(altitude from J to EC)} \cdot EC = \frac{3}{7}y \cdot \frac{1}{2}x \cdot \frac{1}{2}</math> and since <math>xy = 70</math> and we want to find <math>\frac{2}{5}</math> of this, we get our answer is <math>\frac{2}{5} \cdot \frac{3}{7} \cdot \frac{1}{2} \cdot 70 \cdot \frac{1}{2} = \boxed{3}</math>. -SuperJJ
-== See Also ==
+== Solution 5 ==
+<math>[CEF] = \frac{[ABCD]}{4} = \frac{35}{2}</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}{EF} \cdot [CEF] = \frac{3}{5} \cdot \frac{35}{2} = \frac{21}{2}</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>
+<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}{CH} = \frac{HJ}{AC} \cdot \frac{AC}{CH} = \frac{6}{35} \cdot \frac{5}{3} = \frac{2}{7}</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]
+== Solution 6 (Mass Points) ==
+We need one more pair of ratios to fully define our mass point system.  Let's use <math>\triangle AFH \sim \triangle CEH\implies EH:HF = 3:2</math> and now do mass points on <math>\triangle AEG</math>:
+<asy>
+size(250);
+pair A = (0,0), B = (10,0), C = (10,7), D = (0,7);
+pair F = (10/3,0), G = (20/3,0), E = (5,7);
+pair H = intersectionpoint(A--C, E--F);
+pair J = intersectionpoint(A--C, E--G);
+filldraw(A--E--G--cycle, rgb(1,1,1)+opacity(0.3), red+2bp);
+draw(A--B--C--D--cycle);
+draw(A--C);
+draw(E--F);
+draw(E--G);
+draw(A--E, dashed);
+draw(E--B, dashed);
+dot("$A$", A, SW);
+dot("$B$", B, SE);
+dot("$C$", C, NE);
+dot("$D$", D, NW);
+dot("$E$", E, N);
+dot("$F$", F, S);
+dot("$G$", G, S);
+dot("$H$", H, ESE);
+dot("$J$", J, W);
+// mass point labels
+pair mass = A + SW;
+label(scale(0.8)*"$3$", mass, UnFill);
+draw(circle(mass, .4), linewidth(1));
+pair mass = F + S;
+label(scale(0.8)*"$6$", mass, UnFill);
+draw(circle(mass, .4), linewidth(1));
+pair mass = G + SE;
+label(scale(0.8)*"$3$", mass, UnFill);
+draw(circle(mass, .4), linewidth(1));
+pair mass = J + .7*ESE;
+label(scale(0.8)*"$7$", mass, UnFill);
+draw(circle(mass, .4), linewidth(1));
+pair mass = E + N;
+label(scale(0.8)*"$4$", mass, UnFill);
+draw(circle(mass, .4), linewidth(1));
+pair mass = H + .7*WNW;
+label(scale(0.8)*"$10$", mass, UnFill);
+draw(circle(mass, .4), linewidth(1));
+</asy>
+Now it's just a standard "Area Reduction by Ratios"&trade; problem going from:
+<cmath>[ABCD]\xrightarrow[]{\frac{1}{2}}[AEB]\xrightarrow[]{\frac{2}{3}}[AEG]\xrightarrow[]{\frac{3}{7}}[AEJ]\xrightarrow[]{\frac{3}{10}}[HEJ]</cmath>
+or,
+<cmath>70 \cdot \frac{1}{2}\cdot \frac{2}{3}\cdot \frac{3}{7}\cdot \frac{3}{10} = \boxed{\textbf{(C) }3}</cmath>
+~ proloto
+==See also==
 {{AMC12 box|year=2001|num-b=21|num-a=23}}
 {{MAA Notice}}

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
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