Difference between revisions of "2016 AMC 10B Problems/Problem 19"
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− | ==Solution 5== | + | ==Solution 5 (Area)== |
I will calculate <math>\frac{PQ}{QE}</math> using the ratio of the area of <math>\triangle APG</math> and <math>\triangle AEG</math>. | I will calculate <math>\frac{PQ}{QE}</math> using the ratio of the area of <math>\triangle APG</math> and <math>\triangle AEG</math>. | ||
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<cmath>[APG]=[ACG]-[CPG]=\frac{5}{2}-\frac{15}{14}=\frac{35-15}{14}=\frac{20}{14}=\frac{10}{7}</cmath> | <cmath>[APG]=[ACG]-[CPG]=\frac{5}{2}-\frac{15}{14}=\frac{35-15}{14}=\frac{20}{14}=\frac{10}{7}</cmath> | ||
<cmath>[AEG]=\frac{1}{2} \cdot 4 \cdot 3=6</cmath> | <cmath>[AEG]=\frac{1}{2} \cdot 4 \cdot 3=6</cmath> | ||
− | Because <math>\triangle | + | Because <math>\triangle APG</math> and <math>\triangle AEG</math> share the same base <math>AG</math>, the ratio <math>\frac{[AEG]}{[APG]}</math> is equal to the ratio of the altitudes of <math>\triangle APG</math> and <math>\triangle AEG</math> to <math>AG</math>, which is equal to <math>\frac{PQ}{QE}</math>: |
− | <cmath>\frac{PQ}{QE}=\frac{[APG]}{[AEG]}=\frac{\frac{10}{7}}{6}=\frac{10}{42}</cmath> | + | <cmath>\frac{PQ}{QE}=\frac{[APG]}{[AEG]}=\frac{\frac{10}{7}}{6}=\frac{10}{42}=\frac{5}{21}</cmath> |
− | <cmath>\frac{PQ}{PE}=\frac{ | + | <cmath>\frac{PQ}{PE}=\frac{5}{21+5}=\frac{5}{26}</cmath> |
<cmath>\frac{FP}{PE}=\frac{CF}{AE}=\frac{3}{4}</cmath> | <cmath>\frac{FP}{PE}=\frac{CF}{AE}=\frac{3}{4}</cmath> | ||
<cmath>\frac{FP}{EF}=\frac{3}{7}</cmath> | <cmath>\frac{FP}{EF}=\frac{3}{7}</cmath> | ||
<cmath>\frac{PE}{EF}=\frac{4}{7}</cmath> | <cmath>\frac{PE}{EF}=\frac{4}{7}</cmath> | ||
− | <cmath>\frac{PE}{EF} \cdot \frac{PQ}{PE}=\frac{4}{7} \cdot \frac{ | + | <cmath>\frac{PE}{EF} \cdot \frac{PQ}{PE}=\frac{4}{7} \cdot \frac{5}{26}</cmath> |
− | <cmath>\frac{PQ}{EF}=\frac{4}{7} \cdot \frac{ | + | <cmath>\frac{PQ}{EF}=\frac{4}{7} \cdot \frac{5}{26}=\boxed{\textbf{(D)}~\frac{10}{91}}</cmath> |
~isabelchen | ~isabelchen |
Revision as of 12:13, 8 October 2021
Contents
Problem
Rectangle has
and
. Point
lies on
so that
, point
lies on
so that
, and point
lies on
so that
. Segments
and
intersect
at
and
, respectively. What is the value of
?
Solution 1 (Coordinate Geometry)
First, we will define point as the origin. Then, we will find the equations of the following three lines:
,
, and
. The slopes of these lines are
,
, and
, respectively. Next, we will find the equations of
,
, and
. They are as follows:
After drawing in altitudes to
from
,
, and
, we see that
because of similar triangles, and so we only need to find the x-coordinates of
and
.
Finding the intersections of
and
, and
and
gives the x-coordinates of
and
to be
and
. This means that
. Now we can find
Solution 2 (Similar Triangles)
Extend to intersect
at
. Letting
, we have that
Then, notice that and
. Thus, we see that
and
Thus, we see that
Solution 3 (Answer Choices)
Since the opposite sides of a rectangle are parallel and
due to vertical angles,
. Furthermore, the ratio between the side lengths of the two triangles is
. Labeling
and
, we see that
turns out to be equal to
. Since the denominator of
must now be a multiple of 7, the only possible solution in the answer choices is
.
Solution 4 (Area)
I will calculate using similar triangle, and
using area of
and
.
Because
and
share the same base
, the ratio
is equal to the ratio of the altitudes of
and
to
, which is equal to
:
~isabelchen
Solution 5 (Area)
I will calculate using the ratio of the area of
and
.
Because
and
share the same base
, the ratio
is equal to the ratio of the altitudes of
and
to
, which is equal to
:
~isabelchen
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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.