Difference between revisions of "2018 AMC 12A Problems/Problem 20"
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== Solution 1== | == Solution 1== | ||
− | Observe that <math>\triangle{EMI}</math> is isosceles right (<math>M</math> is the midpoint of diameter arc <math>EI</math> since <math>m\angle MEI = m\angle MAI = 45^\circ</math>), so <math>MI=2,MC=\frac{3}{\sqrt{2}}</math>. With <math>\angle{MCI}=45^\circ</math>, we can use Law of Cosines to determine that <math>CI=\frac{3\pm\sqrt{7}}{2}</math>. The same calculations hold for <math>BE</math> also, and since <math>CI<BE</math>, we deduce that <math>CI</math> is the smaller root, giving the answer of <math>\boxed{12}</math>. | + | Observe that <math>\triangle{EMI}</math> is isosceles right (<math>M</math> is the midpoint of diameter arc <math>EI</math> since <math>m\angle MEI = m\angle MAI = 45^\circ</math>), so <math>MI=2,MC=\frac{3}{\sqrt{2}}</math>. With <math>\angle{MCI}=45^\circ</math>, we can use Law of Cosines to determine that <math>CI=\frac{3\pm\sqrt{7}}{2}</math>. The same calculations hold for <math>BE</math> also, and since <math>CI<BE</math>, we deduce that <math>CI</math> is the smaller root, giving the answer of <math>\boxed{\textbf{(D) }12}</math>. |
− | == Solution 2 ( | + | == Solution 2 (Ptolemy) == |
We first claim that <math>\triangle{EMI}</math> is isosceles and right. | We first claim that <math>\triangle{EMI}</math> is isosceles and right. | ||
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Since <math>M</math> is the mid-point of <math>\overline{BC}</math>, it is clear that <math>AM=\frac{3\sqrt{2}}{2}</math>. | Since <math>M</math> is the mid-point of <math>\overline{BC}</math>, it is clear that <math>AM=\frac{3\sqrt{2}}{2}</math>. | ||
− | Now let <math>AE=a</math> and <math>AI=b</math>. By Ptolemy's Theorem, in cyclic quadrilateral <math>AIME</math>, we have <math>2a+2b=6</math>. By Pythagorean Theorem, we have <math>a^2+b^2=8</math>. One can solve the simultaneous system and find <math>b=\frac{3+\sqrt{7}}{2}</math>. Then by deducting the length of <math>\overline{AI}</math> from 3 we get <math>CI=\frac{3-\sqrt{7}}{2}</math>, giving the answer of <math>\boxed{12}</math>. (Surefire2019) | + | Now let <math>AE=a</math> and <math>AI=b</math>. By Ptolemy's Theorem, in cyclic quadrilateral <math>AIME</math>, we have <math>2a+2b=6</math>. By Pythagorean Theorem, we have <math>a^2+b^2=8</math>. One can solve the simultaneous system and find <math>b=\frac{3+\sqrt{7}}{2}</math>. Then by deducting the length of <math>\overline{AI}</math> from 3 we get <math>CI=\frac{3-\sqrt{7}}{2}</math>, giving the answer of <math>\boxed{\textbf{(D) }12}</math>. (Surefire2019) |
− | == Solution 3 ( | + | == Solution 3 (Elementary) == |
− | Like above, notice that <math>\triangle{EMI}</math> is isosceles and right, which means that <math>\dfrac{ME \cdot MI}{2} = 2</math>, so <math>MI^2=4</math> and <math>MI = 2</math>. Then construct <math>\overline{MF}\perp\overline{AB}</math> and <math>\overline{MG}\perp\overline{AC}</math> as well as <math>\overline{MI}</math>. It's clear that <math>MG^2+GI^2 = MI^2</math> by Pythagorean, so knowing that <math>MG = \dfrac{AB}{2} = \dfrac{3}{2}</math> allows one to solve to get <math>GI = \dfrac{\sqrt{7}}{2}</math>. By just looking at the diagram, <math>CI=AC-MF-GI=\dfrac{3-\sqrt{7}}{2}</math>. The answer is thus <math>3+7+2=12</math>. | + | Like above, notice that <math>\triangle{EMI}</math> is isosceles and right, which means that <math>\dfrac{ME \cdot MI}{2} = 2</math>, so <math>MI^2=4</math> and <math>MI = 2</math>. Then construct <math>\overline{MF}\perp\overline{AB}</math> and <math>\overline{MG}\perp\overline{AC}</math> as well as <math>\overline{MI}</math>. It's clear that <math>MG^2+GI^2 = MI^2</math> by Pythagorean, so knowing that <math>MG = \dfrac{AB}{2} = \dfrac{3}{2}</math> allows one to solve to get <math>GI = \dfrac{\sqrt{7}}{2}</math>. By just looking at the diagram, <math>CI=AC-MF-GI=\dfrac{3-\sqrt{7}}{2}</math>. The answer is thus <math>3+7+2=\textbf{(D) }12</math>. |
== Solution 4 (Coordinate Geometry) == | == Solution 4 (Coordinate Geometry) == | ||
− | Let <math>A</math> lie on <math>(0,0)</math>, <math>E</math> on <math>(0,y)</math>, <math>I</math> on <math>(x,0)</math>, and <math>M</math> on <math>(\frac{3}{2},\frac{3}{2})</math>. Since <math>{AIME}</math> is cyclic, <math>\angle EMI</math> (which is opposite of another right angle) must be a right angle; therefore, <math>\vec{ME} \cdot \vec{MI} = <\frac{-3}{2}, y-\frac{3}{2}> \cdot <x-\frac{3}{2}, -\frac{3}{2}> = 0</math>. Compute the dot product to arrive at the relation <math>y=3-x</math>. We can set up another equation involving the area of <math>\triangle EMI</math> using the [[Shoelace Theorem]]. This is <math>2=(\frac{1}{2})[(\frac{3}{2})(y-\frac{3}{2})+(x)(-y)+(x+\frac{3}{2})(\frac{3}{2})]</math>. Multiplying, substituting <math>3-x</math> for <math>y</math>, and simplifying, we get <math>x^2 -3x + \frac{1}{2}=0</math>. Thus, <math>(x,y)=</math> <math>(\frac{3 \pm \sqrt{7}}{2},\frac{3 \mp \sqrt{7}}{2})</math>. But <math>AI>AE</math>, meaning <math>x=AI=\frac{3 + \sqrt{7}}{2} \rightarrow CI = 3-\frac{3 + \sqrt{7}}{2}=\frac{3 - \sqrt{7}}{2}</math>, and the final answer is <math>3+7+2=\boxed{12}</math>. | + | Let <math>A</math> lie on <math>(0,0)</math>, <math>E</math> on <math>(0,y)</math>, <math>I</math> on <math>(x,0)</math>, and <math>M</math> on <math>(\frac{3}{2},\frac{3}{2})</math>. Since <math>{AIME}</math> is cyclic, <math>\angle EMI</math> (which is opposite of another right angle) must be a right angle; therefore, <math>\vec{ME} \cdot \vec{MI} = <\frac{-3}{2}, y-\frac{3}{2}> \cdot <x-\frac{3}{2}, -\frac{3}{2}> = 0</math>. Compute the dot product to arrive at the relation <math>y=3-x</math>. We can set up another equation involving the area of <math>\triangle EMI</math> using the [[Shoelace Theorem]]. This is <math>2=(\frac{1}{2})[(\frac{3}{2})(y-\frac{3}{2})+(x)(-y)+(x+\frac{3}{2})(\frac{3}{2})]</math>. Multiplying, substituting <math>3-x</math> for <math>y</math>, and simplifying, we get <math>x^2 -3x + \frac{1}{2}=0</math>. Thus, <math>(x,y)=</math> <math>(\frac{3 \pm \sqrt{7}}{2},\frac{3 \mp \sqrt{7}}{2})</math>. But <math>AI>AE</math>, meaning <math>x=AI=\frac{3 + \sqrt{7}}{2} \rightarrow CI = 3-\frac{3 + \sqrt{7}}{2}=\frac{3 - \sqrt{7}}{2}</math>, and the final answer is <math>3+7+2=\boxed{\textbf{(D) }12}</math>. |
== Solution 5 (Quick) == | == Solution 5 (Quick) == | ||
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Let <math>CI=AE=x</math>, so <math>AI=3-x</math>. | Let <math>CI=AE=x</math>, so <math>AI=3-x</math>. | ||
− | By Pythagoras on <math>\triangle{EAI}</math> we have <math>x^2+(3-x)^2=EI^2=8</math>, and solve this to get <math>x=CI=\dfrac{3-\sqrt{7}}{2}</math> for a final answer of <math>3+7+2=\boxed{12}</math>. | + | By Pythagoras on <math>\triangle{EAI}</math> we have <math>x^2+(3-x)^2=EI^2=8</math>, and solve this to get <math>x=CI=\dfrac{3-\sqrt{7}}{2}</math> for a final answer of <math>3+7+2=\boxed{\textbf{(D) }12}</math>. |
− | ==Solution 6( | + | |
− | Let <math>CI=a</math>, <math>BE=b</math>. Because opposite angles in a cyclic quadrilateral are supplementary, we have <math>\angle EMI=90^{\circ}</math>. By the law of cosines, we have <math>MI^2=a^2+\frac{9}{4}-\frac{3}{2}a</math>, and <math>ME^2=b^2+\frac{9}{4}-\frac{3}{2}b</math>. Notice that <math>EI=2MO</math>, where <math>O</math> is the origin of the circle mentioned in the problem. Thus <math>\frac{2MO*MO}{2}=2\implies MO=\sqrt{2}, EI=2\sqrt{2}</math>. By the Pythagorean Theorem, we have <math>ME^2+MI^2=EI^2\implies a^2+\frac{9}{4}-\frac{3}{2}a+b^2+\frac{9}{4}-\frac{3}{2}b=(2\sqrt{2})^2=8</math>. By the Pythagorean Theorem, we have <math>AE^2+AI^2=EI^2\implies (3-a)^2+(3-b)^2=(2\sqrt{2})^2=8\implies 18-6a-6b+a^2+b^2=8</math>. Thus we have <math>18-6a-6b+a^2+b^2=a^2+\frac{9}{4}-\frac{3}{2}a+b^2+\frac{9}{4}-\frac{3}{2}b\implies 18-6a-6b=\frac{9}{2}-\frac{3}{2}a-\frac{3}{2}b\implies \frac{27}{2}</math> <math>=\frac{9}{2}a+\frac{9}{2}b\implies a+b=3\implies 3-b=a</math>. We know that <math>(3-a)^2+(3-b)^2=8\implies (3-a)^2+a^2=8\implies 2a^2-6a+9=8\implies 2a^2-6a+1=0\implies a=\frac{6\pm \sqrt{36-8}}{2}=\frac{3\pm\sqrt{7}}{2}</math>. We take the smaller solution because we have <math>AI>AE\implies 3-AI<3-AE\implies CI<CE</math>, and we want <math>CI</math>, not <math>CE</math>, thus <math>CI=\frac{3-\sqrt{7}}{2}</math>. Thus our final answer is <math>3+7+2=\boxed{ | + | ==Solution 6 (Bash)== |
+ | Let <math>CI=a</math>, <math>BE=b</math>. Because opposite angles in a cyclic quadrilateral are supplementary, we have <math>\angle EMI=90^{\circ}</math>. By the law of cosines, we have <math>MI^2=a^2+\frac{9}{4}-\frac{3}{2}a</math>, and <math>ME^2=b^2+\frac{9}{4}-\frac{3}{2}b</math>. Notice that <math>EI=2MO</math>, where <math>O</math> is the origin of the circle mentioned in the problem. Thus <math>\frac{2MO*MO}{2}=2\implies MO=\sqrt{2}, EI=2\sqrt{2}</math>. By the Pythagorean Theorem, we have <math>ME^2+MI^2=EI^2\implies a^2+\frac{9}{4}-\frac{3}{2}a+b^2+\frac{9}{4}-\frac{3}{2}b=(2\sqrt{2})^2=8</math>. By the Pythagorean Theorem, we have <math>AE^2+AI^2=EI^2\implies (3-a)^2+(3-b)^2=(2\sqrt{2})^2=8\implies 18-6a-6b+a^2+b^2=8</math>. Thus we have <math>18-6a-6b+a^2+b^2=a^2+\frac{9}{4}-\frac{3}{2}a+b^2+\frac{9}{4}-\frac{3}{2}b\implies 18-6a-6b=\frac{9}{2}-\frac{3}{2}a-\frac{3}{2}b\implies \frac{27}{2}</math> <math>=\frac{9}{2}a+\frac{9}{2}b\implies a+b=3\implies 3-b=a</math>. We know that <math>(3-a)^2+(3-b)^2=8\implies (3-a)^2+a^2=8\implies 2a^2-6a+9=8\implies 2a^2-6a+1=0\implies a=\frac{6\pm \sqrt{36-8}}{2}=\frac{3\pm\sqrt{7}}{2}</math>. We take the smaller solution because we have <math>AI>AE\implies 3-AI<3-AE\implies CI<CE</math>, and we want <math>CI</math>, not <math>CE</math>, thus <math>CI=\frac{3-\sqrt{7}}{2}</math>. Thus our final answer is <math>3+7+2=\boxed{\textbf{(D) }12}</math>. | ||
-vsamc | -vsamc |
Revision as of 23:27, 17 August 2021
Contents
Problem
Triangle is an isosceles right triangle with
. Let
be the midpoint of hypotenuse
. Points
and
lie on sides
and
, respectively, so that
and
is a cyclic quadrilateral. Given that triangle
has area
, the length
can be written as
, where
,
, and
are positive integers and
is not divisible by the square of any prime. What is the value of
?
Diagram
Solution 1
Observe that is isosceles right (
is the midpoint of diameter arc
since
), so
. With
, we can use Law of Cosines to determine that
. The same calculations hold for
also, and since
, we deduce that
is the smaller root, giving the answer of
.
Solution 2 (Ptolemy)
We first claim that is isosceles and right.
Proof: Construct and
. Since
bisects
, one can deduce that
. Then by AAS it is clear that
and therefore
is isosceles. Since quadrilateral
is cyclic, one can deduce that
. Q.E.D.
Since the area of is 2, we can find that
,
Since is the mid-point of
, it is clear that
.
Now let and
. By Ptolemy's Theorem, in cyclic quadrilateral
, we have
. By Pythagorean Theorem, we have
. One can solve the simultaneous system and find
. Then by deducting the length of
from 3 we get
, giving the answer of
. (Surefire2019)
Solution 3 (Elementary)
Like above, notice that is isosceles and right, which means that
, so
and
. Then construct
and
as well as
. It's clear that
by Pythagorean, so knowing that
allows one to solve to get
. By just looking at the diagram,
. The answer is thus
.
Solution 4 (Coordinate Geometry)
Let lie on
,
on
,
on
, and
on
. Since
is cyclic,
(which is opposite of another right angle) must be a right angle; therefore,
. Compute the dot product to arrive at the relation
. We can set up another equation involving the area of
using the Shoelace Theorem. This is
. Multiplying, substituting
for
, and simplifying, we get
. Thus,
. But
, meaning
, and the final answer is
.
Solution 5 (Quick)
From cyclic we get
and
, so
is an isosceles right triangle.
From we get
.
Notice , because
,
, and
.
Let , so
.
By Pythagoras on we have
, and solve this to get
for a final answer of
.
Solution 6 (Bash)
Let ,
. Because opposite angles in a cyclic quadrilateral are supplementary, we have
. By the law of cosines, we have
, and
. Notice that
, where
is the origin of the circle mentioned in the problem. Thus
. By the Pythagorean Theorem, we have
. By the Pythagorean Theorem, we have
. Thus we have
. We know that
. We take the smaller solution because we have
, and we want
, not
, thus
. Thus our final answer is
.
-vsamc
Video Solution
https://youtu.be/NsQbhYfGh1Q?t=4465
~ pi_is_3.14
See Also
2018 AMC 12A (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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.