Difference between revisions of "2024 AMC 10B Problems/Problem 10"
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==Solution 2== | ==Solution 2== | ||
Let <math>[AFE]=1</math>. Since <math>\triangle AFE\sim\triangle CFB</math> with a scale factor of <math>2</math>, <math>[CFB]=4</math>. The scale factor of <math>2</math> also means that <math>\dfrac{AF}{FC}=\dfrac{1}{2}</math>, therefore since <math>\triangle BCF</math> and <math>\triangle BFA</math> have the same height, <math>[BFA]=2</math>. Since <math>ABCD</math> is a parallelogram, <cmath>[BCA]=[DAC]\implies4+2=1+[CDEF]\implies [CDEF]=5\implies\boxed{\text{(A) }5:4}</cmath> ~Tacos_are_yummy_1 | Let <math>[AFE]=1</math>. Since <math>\triangle AFE\sim\triangle CFB</math> with a scale factor of <math>2</math>, <math>[CFB]=4</math>. The scale factor of <math>2</math> also means that <math>\dfrac{AF}{FC}=\dfrac{1}{2}</math>, therefore since <math>\triangle BCF</math> and <math>\triangle BFA</math> have the same height, <math>[BFA]=2</math>. Since <math>ABCD</math> is a parallelogram, <cmath>[BCA]=[DAC]\implies4+2=1+[CDEF]\implies [CDEF]=5\implies\boxed{\text{(A) }5:4}</cmath> ~Tacos_are_yummy_1 | ||
+ | [[File:2024 AMC 10B 10.png|400px|left]] | ||
==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)== | ==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)== |
Revision as of 08:57, 15 November 2024
Contents
[hide]Problem
Quadrilateral is a parallelogram, and
is the midpoint of the side
. Let
be the intersection of lines
and
. What is the ratio of the area of
quadrilateral
to the area of
?
Solution 1
Let have length
and let the altitude of the parallelogram perpendicular to
have length
.
The area of the parallelogram is and the area of
equals
. Thus, the area of quadrilateral
is
.
We have from that
. Also,
, so the length of the altitude of
from
is twice that of
. This means that the altitude of
is
, so the area of
is
.
Then, the area of quadrilateral equals the area of
minus that of
, which is
. Finally, the ratio of the area of
to the area of triangle
is
, so the answer is
.
Solution 2
Let . Since
with a scale factor of
,
. The scale factor of
also means that
, therefore since
and
have the same height,
. Since
is a parallelogram,
~Tacos_are_yummy_1
Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)
https://youtu.be/QLziG_2e7CY?feature=shared
~ Pi Academy
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=24EZaeAThuE
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.