Difference between revisions of "2018 AMC 8 Problems/Problem 20"
(→Solution 4) |
Ninjaboi2000 (talk | contribs) (→Solution 2) |
||
Line 27: | Line 27: | ||
We can extend it into a parallelogram, so it would equal <math>3a \cdot 3b</math>. The smaller parallelogram is <math>a</math> times <math>2b</math>. The smaller parallelogram is <math>\frac{2}{9}</math> of the larger parallelogram, so the answer would be <math>\frac{2}{9} \cdot 2</math>, since the triangle is <math>\frac{1}{2}</math> of the parallelogram, so the answer is <math>\boxed{(\textbf{A}) \frac{4}{9}}</math>. | We can extend it into a parallelogram, so it would equal <math>3a \cdot 3b</math>. The smaller parallelogram is <math>a</math> times <math>2b</math>. The smaller parallelogram is <math>\frac{2}{9}</math> of the larger parallelogram, so the answer would be <math>\frac{2}{9} \cdot 2</math>, since the triangle is <math>\frac{1}{2}</math> of the parallelogram, so the answer is <math>\boxed{(\textbf{A}) \frac{4}{9}}</math>. | ||
− | By babyzombievillager with credits to many others who helped with the solution: | + | By babyzombievillager with credits to many others who helped with the solution :D |
==Solution 3== | ==Solution 3== |
Revision as of 19:06, 20 October 2020
Contents
Problem 20
In a point is on with and Point is on so that and point is on so that What is the ratio of the area of to the area of
Solution 1
By similar triangles, we have . Similarly, we see that Using this information, we get Then, since , it follows that the . Thus, the answer would be .
Sidenote: denotes the area of triangle . Similarly, denotes the area of figure .
Solution 2
We can extend it into a parallelogram, so it would equal . The smaller parallelogram is times . The smaller parallelogram is of the larger parallelogram, so the answer would be , since the triangle is of the parallelogram, so the answer is .
By babyzombievillager with credits to many others who helped with the solution :D
Solution 3
. We can substitute as and as , where is . Side having, distance , has parts also. And and are and respectfully. You can consider the height of and as and respectfully. The area of is because the area formula for a triangle is or . The area of will be . So the area of will be . The area of parallelogram will be . Parallelogram to . The answer is
Solution 4 (Non-math solution)
If you have little time to calculate, divide DEFC into triangles that are equal to dae. Also cut triangle EFB into triangles similar to DAE. We see that there are 9 total triangles, and 4 of those are occupied by DEFC. Thus, 4/9.
See Also
2018 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.