Difference between revisions of "2005 AMC 10A Problems/Problem 25"
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Let <math>F</math> be on <math>AC</math> such that <math>DE\parallel BF</math> then we have | Let <math>F</math> be on <math>AC</math> such that <math>DE\parallel BF</math> then we have | ||
<cmath>\frac{[ABF]}{[ADE]}=(\frac{AD}{AB})^2=(\frac{19}{25})^2=\frac{361}{625}</cmath> | <cmath>\frac{[ABF]}{[ADE]}=(\frac{AD}{AB})^2=(\frac{19}{25})^2=\frac{361}{625}</cmath> | ||
− | Since <math>\bigtriangleup ADE\sim\bigtriangleup ABF</math> | + | <math></math>\frac{[ADE]}{[DECB]}= |
+ | Since <math>\bigtriangleup ADE\sim\bigtriangleup ABF</math> we have | ||
<cmath>\frac{AD}{AE}=\frac{DB}{EF}\Longrightarrow EF=\frac{84}{19}</cmath> | <cmath>\frac{AD}{AE}=\frac{DB}{EF}\Longrightarrow EF=\frac{84}{19}</cmath> | ||
+ | Thus <math>FC=EC-EF=\frac{448}{19}</math> and | ||
+ | <cmath>\frac{[ABF]}{[BFC]}=\frac{AF}{FC}=\frac{350}{448}</cmath> | ||
+ | Finally, | ||
+ | <math></math>\frac{[ADE]}{[DECB]}=\frac{ | ||
==See also== | ==See also== |
Revision as of 21:54, 28 December 2019
Contents
Problem
In we have , , and . Points and are on and respectively, with and . What is the ratio of the area of triangle to the area of the quadrilateral ?
Solution 1(no trig)
We have that
But , so
Solution 2(no trig)
We can let . Since , . So, . This means that . Thus,
-Conantwiz2023
Solution 3(trig)
Using this formula:
Since the area of is equal to the area of minus the area of ,
.
Therefore, the desired ratio is
Note: was not used in this problem
Solution 4
Let be on such that then we have $$ (Error compiling LaTeX. Unknown error_msg)\frac{[ADE]}{[DECB]}= Since we have Thus and Finally, $$ (Error compiling LaTeX. Unknown error_msg)\frac{[ADE]}{[DECB]}=\frac{
See also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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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