Difference between revisions of "2002 AMC 10A Problems/Problem 20"
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==Solution== | ==Solution== | ||
− | Solution #1: | + | Solution <math>\text{#1}: |
− | Since <math>AG< | + | Since </math>AG<math> and </math>CH<math> are parallel, triangles </math>GAD<math> and </math>HCD<math> are similar. Hence, </math>CH/AG = CD/AD = 1/3<math>. |
− | Since <math>AG< | + | Since </math>AG<math> and </math>JE<math> are parallel, triangles </math>GAF<math> and </math>JEF<math> are similar. Hence, </math>EJ/AG = EF/AF = 1/5<math>. Therefore, </math>CH/EJ = (CH/AG)/(EJ/AG) = (1/3)/(1/5) = \boxed{5/3}<math>. The answer is (D). |
− | Solution #2: | + | Solution \text{#2}: |
− | As <math>\overline{JE}< | + | As </math>\overline{JE}<math> is parallel to </math>\overline{AG}<math>, angles FJE and FGA are congruent. Also, angle F is clearly congruent to itself. From AA similarity, </math>\triangle AGF \sim \triangle EJF<math>; hence </math>\frac {AG}{JE} =5<math>. Similarly, </math>\frac {AG}{HC} = 3<math>. Thus, </math>\frac {HC}{JE} = \boxed{\frac {5}{3}\Rightarrow \text{(D)}}$. |
==See Also== | ==See Also== |
Revision as of 18:24, 28 November 2015
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 .
ABCDEFJGH
Solution
Solution $\text{#1}: Since$ (Error compiling LaTeX. ! You can't use `macro parameter character #' in restricted horizontal mode.)AGCHGADHCDCH/AG = CD/AD = 1/3$.
Since$ (Error compiling LaTeX. ! Missing $ inserted.)AGJEGAFJEFEJ/AG = EF/AF = 1/5CH/EJ = (CH/AG)/(EJ/AG) = (1/3)/(1/5) = \boxed{5/3}$. The answer is (D).
Solution \text{#2}: As$ (Error compiling LaTeX. ! Missing $ inserted.)\overline{JE}\overline{AG}\triangle AGF \sim \triangle EJF\frac {AG}{JE} =5\frac {AG}{HC} = 3\frac {HC}{JE} = \boxed{\frac {5}{3}\Rightarrow \text{(D)}}$.
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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