2002 AMC 10A Problems/Problem 20

Problem

Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$ , dividing it into five segments, each of length 1. Point $G$ is not on line $AF$ . Point $H$ lies on $\overline{GD}$ , and point $J$ lies on $\overline{GF}$ . The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$ .

$\text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 2$

Solution

As $\overline{JE}$ is parallel to $\overline{AG}$ , angles FHD and FGA are congruent. Also, angle F is clearly congruent to itself. From SSS similarity, $\triangle AGF \sim \triangle EJF$ ; hence $\frac {AG}{JE} =5$ . Similarly, $\frac {AG}{HC} = 3$ . Thus, $\frac {HC}{JE} = \boxed{\frac {5}{3}\Rightarrow \text{(D)}}$ .

2002 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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2002 AMC 10A Problems/Problem 20

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