Difference between revisions of "2014 AMC 10A Problems/Problem 16"
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==Solution 2== | ==Solution 2== | ||
− | Let the area of the shaded region be <math>x</math>. Let the other two vertices of the kite be <math>I</math> and <math>J</math> with <math>I</math> closer to <math>AD</math> than <math>J</math>. Note that <math> [ABCD] = [ABF] + [DCH] - x + [ADI] + [BCJ]</math>. The area of <math>ABF</math> is <math>1</math> and the area of <math>DCH</math> is <math>\dfrac{1}{2}</math>. We will solve for the areas of <math>ADI</math> and <math>BCJ</math> in terms of x by noting that the area of each triangle is the length of the perpendicular from <math>I</math> to <math>AD</math> and <math>J</math> to <math>BC</math> respectively. Because the area of <math>x</math> = <math>\dfrac{1}{2} | + | Let the area of the shaded region be <math>x</math>. Let the other two vertices of the kite be <math>I</math> and <math>J</math> with <math>I</math> closer to <math>AD</math> than <math>J</math>. Note that <math> [ABCD] = [ABF] + [DCH] - x + [ADI] + [BCJ]</math>. The area of <math>ABF</math> is <math>1</math> and the area of <math>DCH</math> is <math>\dfrac{1}{2}</math>. We will solve for the areas of <math>ADI</math> and <math>BCJ</math> in terms of x by noting that the area of each triangle is the length of the perpendicular from <math>I</math> to <math>AD</math> and <math>J</math> to <math>BC</math> respectively. Because the area of <math>x</math> = <math>\dfrac{1}{2} \cdot IJ</math> based on the area of a kite formula, <math>\dfrac{ab}{2}</math> for diagonals of length <math>a</math> and <math>b</math>, <math>IJ = 2x</math>. So each perpendicular is length <math>\dfrac{1-2x}{2}</math>. So taking our numbers and plugging them into <math> [ABCD] =[ABF] + [DCH] - x + [ADI] + [BCJ]</math> gives us <math>2 = \dfrac{5}{2} - 3x</math> Solving this equation for <math>x</math> gives us <math> x = \boxed{\textbf{(E)} \: \frac{1}{6}}</math> |
==Solution 3== | ==Solution 3== | ||
− | From the diagram in Solution 1, let <math>e</math> be the height of <math>XHY</math> and <math>f</math> be the height of <math>XFY</math>. It is clear that their sum is <math>1</math> as they are parallel to <math>GD</math>. Let <math>k</math> be the ratio of the sides of the similar triangles <math>XFY</math> and <math>AFB</math>, which are similar because <math>XY</math> is parallel to <math>AB</math> and the triangles share angle <math>F</math>. Then <math>k = f/2</math>, as 2 is the height of <math>AFB</math>. Since <math>XHY</math> and <math>DHC</math> are similar for the same reasons as <math>XFY</math> and <math>AFB</math>, the height of <math>XHY</math> will be equal to the base, like in <math>DHC</math>, making <math>XY = e</math>. However, <math>XY</math> is also the base of <math>XFY</math>, so <math>k = e / AB</math> where <math>AB = 1</math> so <math>k = e</math>. Subbing into <math>k = f/2</math> gives a system of linear equations, <math>e + f = 1</math> and <math>e = f/2</math>. Solving yields <math>e = XY = 1/3</math> and <math>f = 2 | + | From the diagram in Solution 1, let <math>e</math> be the height of <math>XHY</math> and <math>f</math> be the height of <math>XFY</math>. It is clear that their sum is <math>1</math> as they are parallel to <math>GD</math>. Let <math>k</math> be the ratio of the sides of the similar triangles <math>XFY</math> and <math>AFB</math>, which are similar because <math>XY</math> is parallel to <math>AB</math> and the triangles share angle <math>F</math>. Then <math>k = f/2</math>, as 2 is the height of <math>AFB</math>. Since <math>XHY</math> and <math>DHC</math> are similar for the same reasons as <math>XFY</math> and <math>AFB</math>, the height of <math>XHY</math> will be equal to the base, like in <math>DHC</math>, making <math>XY = e</math>. However, <math>XY</math> is also the base of <math>XFY</math>, so <math>k = e / AB</math> where <math>AB = 1</math> so <math>k = e</math>. Subbing into <math>k = f/2</math> gives a system of linear equations, <math>e + f = 1</math> and <math>e = f/2</math>. Solving yields <math>e = XY = 1/3</math> and <math>f = \frac{2}{3}</math>, and since the area of the kite is simply the product of the two diagonals over <math>2</math> and <math>HF = 1</math>, our answer is <math>\frac{\frac{1}{3} \cdot 1}{2} = \boxed{\textbf{(E)} \: \dfrac16}</math>. |
==Solution 4== | ==Solution 4== | ||
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Lemma: The distance from line segment <math>JI</math> to <math>H</math> is half the distance from <math>JI</math> to <math>F</math> | Lemma: The distance from line segment <math>JI</math> to <math>H</math> is half the distance from <math>JI</math> to <math>F</math> | ||
− | Proof: Drop perpendiculars of triangles <math>HJI</math> and <math>JIF</math> to line <math>JI</math>, and let the point of intersection be <math>Q</math>. Note that <math>HJI</math> and <math>JIF</math> are similar to <math>HDC</math> and <math>ABF</math>, respectively. Now, the ratio of <math>DC</math> to <math>HF</math> is <math>1:1</math>, which shows that the ratio of <math>JI</math> to <math>HQ</math> is <math>1:1</math>, because of similar triangles as described above. Similarly, the ratio of <math>JI</math> to <math>FQ</math> is <math>1:2</math>. Since these two triangles contain the same base, <math>JI</math>, the ratio of <math>HQ:FQ = 1:2</math>. | + | Proof: Drop perpendiculars of triangles <math>HJI</math> and <math>JIF</math> to line <math>JI</math>, and let the point of intersection be <math>Q</math>. Note that <math>HJI</math> and <math>JIF</math> are similar to <math>HDC</math> and <math>ABF</math>, respectively. Now, the ratio of <math>DC</math> to <math>HF</math> is <math>1:1</math>, which shows that the ratio of <math>JI</math> to <math>HQ</math> is <math>1:1</math>, because of similar triangles as described above. Similarly, the ratio of <math>JI</math> to <math>FQ</math> is <math>1:2</math>. Since these two triangles contain the same base, <math>JI</math>, the ratio of <math>HQ:FQ = 1:2</math>. |
− | Because kite <math>HJFI</math> is orthodiagonal, we multiply <math> | + | Because kite <math>HJFI</math> is orthodiagonal, we multiply <math>\frac{1\cdot\tfrac{1}{3}}{2} = \boxed{\textbf{(E)} \: \frac{1}{6}}</math> |
~Lemma proof by sakshamsethi | ~Lemma proof by sakshamsethi |
Revision as of 11:05, 7 September 2021
Problem
In rectangle ,
,
, and points
,
, and
are midpoints of
,
, and
, respectively. Point
is the midpoint of
. What is the area of the shaded region?
Solution 1
Denote . Then
. Let the intersection of
and
be
, and the intersection of
and
be
. Then we want to find the coordinates of
so we can find
. From our points, the slope of
is
, and its
-intercept is just
. Thus the equation for
is
. We can also quickly find that the equation of
is
. Setting the equations equal, we have
. Because of symmetry, we can see that the distance from
to
is also
, so
. Now the area of the kite is simply the product of the two diagonals over
. Since the length
, our answer is
.
Solution 2
Let the area of the shaded region be . Let the other two vertices of the kite be
and
with
closer to
than
. Note that
. The area of
is
and the area of
is
. We will solve for the areas of
and
in terms of x by noting that the area of each triangle is the length of the perpendicular from
to
and
to
respectively. Because the area of
=
based on the area of a kite formula,
for diagonals of length
and
,
. So each perpendicular is length
. So taking our numbers and plugging them into
gives us
Solving this equation for
gives us
Solution 3
From the diagram in Solution 1, let be the height of
and
be the height of
. It is clear that their sum is
as they are parallel to
. Let
be the ratio of the sides of the similar triangles
and
, which are similar because
is parallel to
and the triangles share angle
. Then
, as 2 is the height of
. Since
and
are similar for the same reasons as
and
, the height of
will be equal to the base, like in
, making
. However,
is also the base of
, so
where
so
. Subbing into
gives a system of linear equations,
and
. Solving yields
and
, and since the area of the kite is simply the product of the two diagonals over
and
, our answer is
.
Solution 4
Let the unmarked vertices of the shaded area be labeled and
, with
being closer to
than
. Noting that kite
can be split into triangles
and
.
Lemma: The distance from line segment to
is half the distance from
to
Proof: Drop perpendiculars of triangles and
to line
, and let the point of intersection be
. Note that
and
are similar to
and
, respectively. Now, the ratio of
to
is
, which shows that the ratio of
to
is
, because of similar triangles as described above. Similarly, the ratio of
to
is
. Since these two triangles contain the same base,
, the ratio of
.
Because kite is orthodiagonal, we multiply
~Lemma proof by sakshamsethi
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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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