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

Revision as of 20:59, 15 August 2021

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

$[asy] unitsize(0.5cm); defaultpen(0.8); pair A=(0,0), B=(10,0), C=(10,7), D=(0,7), E=(C+D)/2, F=(2*A+B)/3, G=(A+2*B)/3; pair H = intersectionpoint(A--C,E--F); pair J = intersectionpoint(A--C,E--G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$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 $\boxed{3}$ .

@@ Line 66: / Line 66: @@
 === 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 <math>7</math> and <math>10</math> (or any other two positive numbers that multiply to 70). We can find <math>H</math> and <math>J</math> by intersecting lines, and then we calculate the area of <math>EHJ</math> using shoelace formula. This yields <math>\boxed{3}</math>.
+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 <math>7</math> and <math>10</math> (or any other two positive real numbers that multiply to 70). We can find <math>H</math> and <math>J</math> by intersecting lines, and then we calculate the area of <math>EHJ</math> using shoelace formula. This yields <math>\boxed{3}</math>.
 == 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
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