Difference between revisions of "2003 AMC 12A Problems/Problem 17"
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Square <math>ABCD</math> has sides of length <math>4</math>, and <math>M</math> is the midpoint of <math>\overline{CD}</math>. A circle with radius <math>2</math> and center <math>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance from <math>P</math> to <math>\overline{AD}</math>? | Square <math>ABCD</math> has sides of length <math>4</math>, and <math>M</math> is the midpoint of <math>\overline{CD}</math>. A circle with radius <math>2</math> and center <math>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance from <math>P</math> to <math>\overline{AD}</math>? | ||
− | + | <asy> | |
+ | pair A,B,C,D,M,P; | ||
+ | D=(0,0); | ||
+ | C=(10,0); | ||
+ | B=(10,10); | ||
+ | A=(0,10); | ||
+ | M=(5,0); | ||
+ | P=(8,4); | ||
+ | dot(M); | ||
+ | dot(P); | ||
+ | draw(A--B--C--D--cycle,linewidth(0.7)); | ||
+ | draw((5,5)..D--C..cycle,linewidth(0.7)); | ||
+ | draw((7.07,2.93)..B--A--D..cycle,linewidth(0.7)); | ||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,NE); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,SW); | ||
+ | label("$M$",M,S); | ||
+ | label("$P$",P,N); | ||
+ | </asy> | ||
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}</math> | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}</math> | ||
Line 34: | Line 53: | ||
Thus <math>y = 0</math> and <math>y = \frac{8}{5}</math> making <math>x = 0</math> and <math>x = \frac{16}{5}</math>. | Thus <math>y = 0</math> and <math>y = \frac{8}{5}</math> making <math>x = 0</math> and <math>x = \frac{16}{5}</math>. | ||
− | The first value of <math>0</math> is obviously referring to the x-coordinate of the point where the circles intersect at the origin, <math>D</math>, so the second value must be referring to the x coordinate of <math>P</math>. Since <math>\overline{AD}</math> is the y-axis, the distance to it from <math>P</math> is the same as the x-value of the coordinate of <math>P</math>, so the distance from <math>P</math> to <math>\overline{AD}</math> is <math>\frac{16}{5} \Rightarrow B</math> | + | The first value of <math>0</math> is obviously referring to the x-coordinate of the point where the circles intersect at the origin, <math>D</math>, so the second value must be referring to the x coordinate of <math>P</math>. Since <math>\overline{AD}</math> is the y-axis, the distance to it from <math>P</math> is the same as the x-value of the coordinate of <math>P</math>, so the distance from <math>P</math> to <math>\overline{AD}</math> is <math>\frac{16}{5} \Rightarrow \boxed{B}.</math> |
− | ==Solution 2== | + | -mathbeast9 |
+ | |||
+ | (JK lol) | ||
+ | |||
+ | == Solution 2 == | ||
+ | <math>APMD</math> obviously forms a kite. Let the intersection of the diagonals be <math>E</math>. <math>AE+EM=AM=2\sqrt{5}</math> Let <math>AE=x</math>. Then, <math>EM=2\sqrt{5}-x</math>. | ||
+ | |||
+ | |||
+ | By Pythagorean Theorem, <math>DE^2=4^2-AE^2=2^2-EM^2</math>. Thus, <math>16-x^2=4-(2\sqrt{5}-x)^2</math>. Simplifying, <math>x=\frac{8}{\sqrt{5}}</math>. By Pythagoras again, <math>DE=\frac{4}{\sqrt{5}}</math>. Then, the area of <math>ADP</math> is <math>DE\cdot AE=\frac{32}{5}</math>. | ||
+ | |||
+ | |||
+ | Using <math>4</math> instead as the base, we can drop a altitude from P. <math>\frac{32}{5}=\frac{bh}{2}</math>. <math>\frac{32}{5}=\frac{4h}{2}</math>. Thus, the horizontal distance is <math>\frac{16}{5} \implies \boxed{\textbf{(B)}\frac{16}{5}}</math> | ||
+ | |||
+ | |||
+ | |||
+ | ~BJHHar | ||
+ | |||
+ | == Solution 3 == | ||
Note that <math>P</math> is merely a reflection of <math>D</math> over <math>AM</math>. Call the intersection of <math>AM</math> and <math>DP</math> <math>X</math>. Drop perpendiculars from <math>X</math> and <math>P</math> to <math>AD</math>, and denote their respective points of intersection by <math>J</math> and <math>K</math>. We then have <math>\triangle DXJ\sim\triangle DPK</math>, with a scale factor of 2. Thus, we can find <math>XJ</math> and double it to get our answer. With some analytical geometry, we find that <math>XJ=\frac{8}{5}</math>, implying that <math>PK=\frac{16}{5}</math>. | Note that <math>P</math> is merely a reflection of <math>D</math> over <math>AM</math>. Call the intersection of <math>AM</math> and <math>DP</math> <math>X</math>. Drop perpendiculars from <math>X</math> and <math>P</math> to <math>AD</math>, and denote their respective points of intersection by <math>J</math> and <math>K</math>. We then have <math>\triangle DXJ\sim\triangle DPK</math>, with a scale factor of 2. Thus, we can find <math>XJ</math> and double it to get our answer. With some analytical geometry, we find that <math>XJ=\frac{8}{5}</math>, implying that <math>PK=\frac{16}{5}</math>. | ||
− | ==Solution | + | == Solution 4 == |
As in Solution 2, draw in <math>DP</math> and <math>AM</math> and denote their intersection point <math>X</math>. Next, drop a perpendicular from <math>P</math> to <math>AD</math> and denote the foot as <math>Z</math>. <math>AP \cong AD</math> as they are both radii and similarly <math>DM \cong MP</math> so <math>APMD</math> is a kite and <math>DX \perp XM</math> by a well-known theorem. | As in Solution 2, draw in <math>DP</math> and <math>AM</math> and denote their intersection point <math>X</math>. Next, drop a perpendicular from <math>P</math> to <math>AD</math> and denote the foot as <math>Z</math>. <math>AP \cong AD</math> as they are both radii and similarly <math>DM \cong MP</math> so <math>APMD</math> is a kite and <math>DX \perp XM</math> by a well-known theorem. | ||
Pythagorean theorem gives us <math>AM=2 \sqrt{5}</math>. Clearly <math>\triangle XMD \sim \triangle XDA \sim \triangle DMA \sim \triangle ZDP</math> by angle-angle and <math>\triangle XMD \cong \triangle XMP</math> by Hypotenuse Leg. | Pythagorean theorem gives us <math>AM=2 \sqrt{5}</math>. Clearly <math>\triangle XMD \sim \triangle XDA \sim \triangle DMA \sim \triangle ZDP</math> by angle-angle and <math>\triangle XMD \cong \triangle XMP</math> by Hypotenuse Leg. | ||
Manipulating similar triangles gives us <math>PZ=\frac{16}{5}</math> | Manipulating similar triangles gives us <math>PZ=\frac{16}{5}</math> | ||
+ | |||
+ | == Solution 5 == | ||
+ | Using the double-angle formula for sine, what we need to find is <math>AP\cdot \sin(DAP) = AP\cdot 2\sin( DAM) \cos(DAM) = 4\cdot 2\cdot \frac{2}{\sqrt{20}}\cdot\frac{4}{\sqrt{20}} = \frac{16}{5}</math>. | ||
+ | |||
+ | ==Solution 6(LoC)== | ||
+ | We use the Law of Cosines: | ||
+ | |||
+ | <math>32-32 \cos \theta = 8 + 8 \cos \theta </math> | ||
+ | |||
+ | <math>\frac{3}{5} = \cos \theta </math> | ||
+ | |||
+ | <math>2 + 2*\frac{3}{5} = \frac{16}{5}</math> | ||
+ | |||
+ | == Solution 7 == | ||
+ | |||
+ | Let <math>H</math> be the foot of the perpendicular from <math>P</math> to <math>CD</math>, and let <math>HD = x</math>. Then we have <math>HC = 4-x</math>, and <math>PH = 4 - \sqrt{16 - x^2}</math>. Since <math>\triangle DHP \sim \triangle PHC</math>, we have <math>HP^2 = DH \cdot HC</math>, or <math>-x^2 + 4x = 16 - 8\sqrt{16-x^2}</math>. Solving gives <math>x = \frac{16}{5}</math>. | ||
+ | |||
+ | ==Solution 8== | ||
+ | |||
+ | <asy> | ||
+ | size(8cm, 8cm); | ||
+ | pair A,B,C,D,M,P,Q,R; | ||
+ | D = (0,0); | ||
+ | C = (10,0); | ||
+ | B = (10,10); | ||
+ | A = (0,10); | ||
+ | M = (5,0); | ||
+ | P = (8,4); | ||
+ | Q = (D+P)/2; | ||
+ | R = (0,4); | ||
+ | dot(M); | ||
+ | dot(P); | ||
+ | draw(A--B--C--D--cycle,linewidth(0.7)); | ||
+ | draw((5,5)..D--C..cycle,linewidth(0.7)); | ||
+ | draw((7.07,2.93)..B--A--D..cycle,linewidth(0.7)); | ||
+ | draw(A--M,linewidth(0.7)); | ||
+ | draw(A--P,linewidth(0.7)); | ||
+ | draw(D--P,linewidth(0.7)); | ||
+ | draw(R--P,linewidth(0.7)); | ||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,NE); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,SW); | ||
+ | label("$M$",M,S); | ||
+ | label("$P$",P,N); | ||
+ | label("$Q$",Q,W); | ||
+ | label("$R$",R,W); | ||
+ | |||
+ | draw(rightanglemark(M, Q, P), linewidth(.5)); | ||
+ | </asy> | ||
+ | |||
+ | Draw <math>AM</math>, <math>DP</math>, and <math>PR</math>. <math>PR</math> is parallel with <math>CD</math> | ||
+ | |||
+ | <math>[AMD] = \frac12 \cdot AD \cdot DM = 4</math>, <math>AM = \sqrt{AD^2 + DM^2} = 2 \sqrt{5}</math> | ||
+ | |||
+ | <math>\triangle ADQ \sim \triangle AMD</math> by <math>AA</math>, <math>[ADQ] = [AMD] \cdot \left( \frac{AD}{AM} \right) ^2 = 4 \cdot \left( \frac{2 \sqrt{5}}{5} \right)^2 = \frac{16}{5}</math> | ||
+ | |||
+ | <math>\triangle ADQ \cong \triangle APQ</math>, <math>[APD] = 2 \cdot [ADQ] = 2 \cdot \frac{16}{5} = \frac{32}{5}</math> | ||
+ | |||
+ | <math>PR = \frac{2 \cdot [APD]}{AD} = \frac{2 \cdot \frac{32}{5}}{4} = \boxed{\textbf{(B) } \frac{16}{5} }</math> | ||
+ | |||
+ | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ||
== See Also == | == See Also == | ||
− | + | {{AMC12 box|year=2003|ab=A|num-b=16|num-a=18}} | |
− | |||
− | |||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:35, 29 March 2023
Contents
Problem
Square has sides of length
, and
is the midpoint of
. A circle with radius
and center
intersects a circle with radius
and center
at points
and
. What is the distance from
to
?
Solution 1
Let be the origin.
is the point
and
is the point
. We are given the radius of the quarter circle and semicircle as
and
, respectively, so their equations, respectively, are:
Subtract the second equation from the first:
Then substitute:
Thus and
making
and
.
The first value of is obviously referring to the x-coordinate of the point where the circles intersect at the origin,
, so the second value must be referring to the x coordinate of
. Since
is the y-axis, the distance to it from
is the same as the x-value of the coordinate of
, so the distance from
to
is
-mathbeast9
(JK lol)
Solution 2
obviously forms a kite. Let the intersection of the diagonals be
.
Let
. Then,
.
By Pythagorean Theorem, . Thus,
. Simplifying,
. By Pythagoras again,
. Then, the area of
is
.
Using instead as the base, we can drop a altitude from P.
.
. Thus, the horizontal distance is
~BJHHar
Solution 3
Note that is merely a reflection of
over
. Call the intersection of
and
. Drop perpendiculars from
and
to
, and denote their respective points of intersection by
and
. We then have
, with a scale factor of 2. Thus, we can find
and double it to get our answer. With some analytical geometry, we find that
, implying that
.
Solution 4
As in Solution 2, draw in and
and denote their intersection point
. Next, drop a perpendicular from
to
and denote the foot as
.
as they are both radii and similarly
so
is a kite and
by a well-known theorem.
Pythagorean theorem gives us . Clearly
by angle-angle and
by Hypotenuse Leg.
Manipulating similar triangles gives us
Solution 5
Using the double-angle formula for sine, what we need to find is .
Solution 6(LoC)
We use the Law of Cosines:
Solution 7
Let be the foot of the perpendicular from
to
, and let
. Then we have
, and
. Since
, we have
, or
. Solving gives
.
Solution 8
Draw ,
, and
.
is parallel with
,
by
,
,
See Also
2003 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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.