Difference between revisions of "2002 AMC 10A Problems/Problem 20"
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Since <math>\overline{AG}</math> and <math>\overline{CH}</math> are parallel, from AA similarity, triangles <math>GAD</math> and <math>HCD</math> are similar. Hence, <math>\frac{CH}{AG} = \frac{CD}{AD} = \frac{1}{3}</math>. | Since <math>\overline{AG}</math> and <math>\overline{CH}</math> are parallel, from AA similarity, triangles <math>GAD</math> and <math>HCD</math> are similar. Hence, <math>\frac{CH}{AG} = \frac{CD}{AD} = \frac{1}{3}</math>. | ||
− | Since <math>\overline{AG}</math> and <math>\overline{JE}</math> are parallel, from AA similarity, triangles <math>GAF</math> and <math>JEF</math> are similar. Hence, <math>\frac{EJ}{AG} = \frac{EF}{AF} = \frac{1}{5}</math>. Therefore, <math>\frac{CH}{EJ} = (\frac{CH}{AG}) | + | Since <math>\overline{AG}</math> and <math>\overline{JE}</math> are parallel, from AA similarity, triangles <math>GAF</math> and <math>JEF</math> are similar. Hence, <math>\frac{EJ}{AG} = \frac{EF}{AF} = \frac{1}{5}</math>. Therefore, <math>\frac{CH}{EJ} = (\frac{CH}{AG})\div(\frac{EJ}{AG}) = (\frac{1}{3})\div(\frac{1}{5}) = \boxed{\frac{5}{3}}</math>. The answer is <math>\boxed{(D)}</math>. |
==Solution 2== | ==Solution 2== |
Revision as of 21:35, 28 December 2015
Contents
Problem
Points and lie, in that order, on , dividing it into five segments, each of length 1. Point is not on line . Point lies on , and point lies on . The line segments and are parallel. Find .
Solution 1
Since and are parallel, from AA similarity, triangles and are similar. Hence, .
Since and are parallel, from AA similarity, triangles and are similar. Hence, . Therefore, . The answer is .
Solution 2
As is parallel to , angles FJE and FGA are congruent. Also, angle F is clearly congruent to itself. From AA similarity, ; hence . Similarly, . Thus, .
See Also
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 |
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