Difference between revisions of "2018 AMC 8 Problems/Problem 20"
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In <math>\triangle ABC,</math> a point <math>E</math> is on <math>\overline{AB}</math> with <math>AE=1</math> and <math>EB=2.</math> Point <math>D</math> is on <math>\overline{AC}</math> so that <math>\overline{DE} \parallel \overline{BC}</math> and point <math>F</math> is on <math>\overline{BC}</math> so that <math>\overline{EF} \parallel \overline{AC}.</math> What is the ratio of the area of <math>CDEF</math> to the area of <math>\triangle ABC?</math> | In <math>\triangle ABC,</math> a point <math>E</math> is on <math>\overline{AB}</math> with <math>AE=1</math> and <math>EB=2.</math> Point <math>D</math> is on <math>\overline{AC}</math> so that <math>\overline{DE} \parallel \overline{BC}</math> and point <math>F</math> is on <math>\overline{BC}</math> so that <math>\overline{EF} \parallel \overline{AC}.</math> What is the ratio of the area of <math>CDEF</math> to the area of <math>\triangle ABC?</math> | ||
Latest revision as of 22:07, 11 January 2025
Contents
[hide]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
Then, since
, it follows that the
.
Sidenote:
Solution 2
Let and
the height of
. We can extend
to form a parallelogram, which 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
.
Video Solution (CREATIVE ANALYSIS!!!)
~Education, the Study of Everything
Video Solution (Meta-Solving Technique)
https://youtu.be/GmUWIXXf_uk?t=1541
~ pi_is_3.14
Video Solution 2
~savannahsolver
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=TpsuRedYOiM&t=250s
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 |
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