Difference between revisions of "2021 AIME I Problems/Problem 11"
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==Solution 4 (Symmetry)== | ==Solution 4 (Symmetry)== | ||
− | Lemma 1 | + | ===Lemma 1=== |
+ | [[File:2021 AIME I 11.jpg|500px|right]] | ||
+ | In the triangle <math>ABC</math>, the points <math>B'</math> and <math>C'</math> are the bases of the heights dropped from the vertices <math>B</math> and <math>C</math>, respectively. <math>\angle A = \alpha</math>. Then <math>B'C' = BC \cos\alpha,</math> <math>\alpha < \frac{\pi}{2}</math> and <math>B'C' = BC \cos (\pi – \alpha), \alpha >\frac{\pi}{2}.</math> | ||
− | + | <i><b>Proof</b></i> | |
− | |||
Denote the orthocenter by <math>A'</math>. Quadrilateral <math>B'C'BC</math> is inscribed in a circle with diameter <math>BC</math>, so the marked <math>\angle B = \angle B'.</math> | Denote the orthocenter by <math>A'</math>. Quadrilateral <math>B'C'BC</math> is inscribed in a circle with diameter <math>BC</math>, so the marked <math>\angle B = \angle B'.</math> | ||
Line 173: | Line 174: | ||
<math>A'C' : AC = \cos (\pi – \alpha).</math> So <math>B'C' : BC = \cos (\pi – \alpha).</math> | <math>A'C' : AC = \cos (\pi – \alpha).</math> So <math>B'C' : BC = \cos (\pi – \alpha).</math> | ||
− | |||
− | |||
+ | ===Lemma 2=== | ||
+ | [[File:2021 AIME I 11c.jpg|400px|right]] | ||
Given an inscribed quadrilateral <math>ABCD</math> with sides <math>AB = a, BC = b, CD = c,</math> and <math>DA = d.</math> Prove that the <math>\angle \theta < \frac{\pi}{2}</math> between the diagonals is given by | Given an inscribed quadrilateral <math>ABCD</math> with sides <math>AB = a, BC = b, CD = c,</math> and <math>DA = d.</math> Prove that the <math>\angle \theta < \frac{\pi}{2}</math> between the diagonals is given by | ||
<cmath>\begin{align*}2(ac + bd) \cos \theta = {|d^2 – c^2 + b^2 – a^2|}.\end{align*}</cmath> | <cmath>\begin{align*}2(ac + bd) \cos \theta = {|d^2 – c^2 + b^2 – a^2|}.\end{align*}</cmath> | ||
− | Proof | + | <i><b>Proof</b></i> |
+ | |||
+ | Let the point <math>B'</math> be symmetric to <math>B</math> with respect to the perpendicular bisector <math>AC.</math> Then the quadrilateral <math>AB'CD</math> is an inscribed one, <math>AB' = b, B'C = a.</math> | ||
<cmath> 2 \angle AEB = \overset{\Large\frown} {AB} + \overset{\Large\frown} {CD}.</cmath> | <cmath> 2 \angle AEB = \overset{\Large\frown} {AB} + \overset{\Large\frown} {CD}.</cmath> | ||
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<cmath>\angle AEB = \angle B'AD.</cmath> | <cmath>\angle AEB = \angle B'AD.</cmath> | ||
We apply the Law of Cosines to <math>\triangle AB'D</math> and <math>\triangle CB'D</math>: | We apply the Law of Cosines to <math>\triangle AB'D</math> and <math>\triangle CB'D</math>: | ||
− | <cmath>\begin{align*} B'D^2 = AD^2 + AB'^2 – 2 AD \cdot AB' \cos \theta = CD^2 + CB'^2 + 2 CD \cdot CB' \cos \theta,\end{align*}</cmath> | + | <cmath>\begin{align*} B'D^2 = AD^2 + AB'^2 – 2 AD \cdot AB' \cos \theta,\end{align*}</cmath> |
− | <cmath>\begin{align*} d^2 + b^2 – 2 | + | <cmath>\begin{align*} B'D^2 = CD^2 + CB'^2 + 2 CD \cdot CB' \cos \theta,\end{align*}</cmath> |
+ | <cmath>\begin{align*} d^2 + b^2 – 2 bd \cos \theta = c^2 + a^2 + 2ac \cos \theta,\end{align*}</cmath> | ||
<cmath>\begin{align*} 2(ac + bd) \cos \theta = |d^2 – c^2 + b^2 – a^2|.\end{align*}</cmath> | <cmath>\begin{align*} 2(ac + bd) \cos \theta = |d^2 – c^2 + b^2 – a^2|.\end{align*}</cmath> | ||
− | |||
− | Solution: | + | <i><b>Solution</b></i> |
− | In accordance with Lemma 1, the ratios of pairs of one-color segments are the same and equal to \cos \theta, where \theta is the acute angle between the diagonals. | + | [[File:2021 AIME I 11b.jpg|450px|right]] |
− | <cmath>s = A'B' + B'C' + C'D' + D'A' = (AB + BC + CD + DA)\cos \theta = (a + b + c + d)\cos \theta = 22\cos \theta.</cmath> | + | In accordance with <b>Lemma 1,</b> the ratios of pairs of one-color segments are the same and equal to <math>\cos \theta,</math> where <math>\theta</math> is the acute angle between the diagonals. |
+ | <cmath>s = A'B' + B'C' + C'D' + D'A'=</cmath> | ||
+ | <cmath> = (AB + BC + CD + DA)\cos \theta =</cmath> | ||
+ | <cmath>= (a + b + c + d)\cos \theta = 22\cos \theta.</cmath> | ||
− | In accordance with Lemma 2, | + | In accordance with <b>Lemma 2,</b> |
<cmath>\begin{align*} 2(ac + bd)\cos \theta = |d^2 – c^2 + b^2 – a^2|.\end{align*}</cmath> | <cmath>\begin{align*} 2(ac + bd)\cos \theta = |d^2 – c^2 + b^2 – a^2|.\end{align*}</cmath> | ||
<cmath>2 \cdot 59 \cos \theta = |13 + 9|.</cmath> | <cmath>2 \cdot 59 \cos \theta = |13 + 9|.</cmath> | ||
<cmath>s = 22\cos \theta = \frac{22 \cdot 11}{59} = \frac{242}{59}.</cmath> | <cmath>s = 22\cos \theta = \frac{22 \cdot 11}{59} = \frac{242}{59}.</cmath> | ||
− | |||
− | |||
~vvsss, www.deoma-cmd.ru | ~vvsss, www.deoma-cmd.ru |
Revision as of 15:33, 30 May 2022
Contents
Problem
Let be a cyclic quadrilateral with
and
Let
and
be the feet of the perpendiculars from
and
respectively, to line
and let
and
be the feet of the perpendiculars from
and
respectively, to line
The perimeter of
is
where
and
are relatively prime positive integers. Find
Diagram
~MRENTHUSIASM
Solution 1 (Cyclic Quadrilaterals, Similar Triangles, and Trigonometry)
This solution refers to the Diagram section.
By the Converse of the Inscribed Angle Theorem, if distinct points and
lie on the same side of
(but not on
itself) for which
then
and
are cyclic. From the Converse of the Inscribed Angle Theorem, quadrilaterals
and
are all cyclic.
Suppose and
intersect at
and let
It follows that
and
We obtain the following diagram:
In every cyclic quadrilateral, each pair of opposite angles is supplementary. So, we have
(both supplementary to
) and
(both supplementary to
), from which
by AA, with the ratio of similitude
Similarly, we have
(both supplementary to
) and
(both supplementary to
), from which
by AA, with the ratio of similitude
We apply the Transitive Property to
and
- We get
so
by SAS, with the ratio of similitude
- We get
so
by SAS, with the ratio of similitude
From and
the perimeter of
is
Two solutions follow from here:
Solution 1.1 (Law of Cosines)
Note that holds for all
We apply the Law of Cosines to
and
respectively:
We subtract
from
Finally, substituting this result into
gives
from which the answer is
~MRENTHUSIASM (credit given to Math Jams's 2021 AIME I Discussion)
Solution 1.2 (Area Formulas)
Let the brackets denote areas.
We find in two different ways:
- Note that
holds for all
By area addition, we get
- By Brahmagupta's Formula, we get
where
is the semiperimeter of
Equating the expressions for we have
so
Since
we have
It follows that
Finally, substituting this result into
gives
from which the answer is
~MRENTHUSIASM (credit given to Leonard my dude)
Remark (Ptolemy's Theorem)
In we have
~MRENTHUSIASM
Solution 2 (Finding cos x)
The angle between diagonals satisfies
(see https://en.wikipedia.org/wiki/Cyclic_quadrilateral#Angle_formulas).
Thus,
That is,
or
. Thus,
or
. So,
In this context,
. Thus,
. The perimeter of
is
and the answer is
.
~y.grace.yu
Solution 3 (Pythagorean Theorem)
We assume that the two quadrilateral mentioned in the problem are similar (due to both of them being cyclic). Note that by Ptolemy’s, one of the diagonals has length [I don't believe this is correct... are the two diagonals of
necessarily congruent? -peace09] WLOG we focus on diagonal
To find the diagonal of the inner quadrilateral, we drop the altitude from
and
and calculate the length of
Let
be
(Thus
By Pythagorean theorem, we have
Now let
be
(thus making
). Similarly, we have
We see that
, the scaled down diagonal is just
which is
times our original diagonal
implying a scale factor of
Thus, due to perimeters scaling linearly, the perimeter of the new quadrilateral is simply
making our answer
~fidgetboss_4000
Solution 4 (Symmetry)
Lemma 1
In the triangle , the points
and
are the bases of the heights dropped from the vertices
and
, respectively.
. Then
and
Proof
Denote the orthocenter by . Quadrilateral
is inscribed in a circle with diameter
, so the marked
If the
the similarity coefficient is
So
If the
the similarity coefficient is
So
Lemma 2
Given an inscribed quadrilateral with sides
and
Prove that the
between the diagonals is given by
Proof
Let the point be symmetric to
with respect to the perpendicular bisector
Then the quadrilateral
is an inscribed one,
We apply the Law of Cosines to
and
:
Solution
In accordance with Lemma 1, the ratios of pairs of one-color segments are the same and equal to where
is the acute angle between the diagonals.
In accordance with Lemma 2,
~vvsss, www.deoma-cmd.ru
See Also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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