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Difference between revisions of "2003 AMC 12A Problems/Problem 17"

(→‎Solution 6(LoC))
(→‎Solution 6(LoC))
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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>.
 
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)==
 
==Solution 6(LoC)==
−
We use LoC:
+
We use the Law of Cosines:
  
 
<math>32-32 \cos \theta = 8 + 8 \cos \theta </math>
 
<math>32-32 \cos \theta = 8 + 8 \cos \theta </math>

Revision as of 11:41, 15 October 2020

Problem

Square $ABCD$ has sides of length $4$, and $M$ is the midpoint of $\overline{CD}$. A circle with radius $2$ and center $M$ intersects a circle with radius $4$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$?

[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]

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}$

Solution 1

Let $D$ be the origin. $A$ is the point $(0,4)$ and $M$ is the point $(2,0)$. We are given the radius of the quarter circle and semicircle as $4$ and $2$, respectively, so their equations, respectively, are:

$x^2 + (y-4)^2 = 4^2$

$(x-2)^2 + y^2 = 2^2$

Subtract the second equation from the first:

$x^2 + (y - 4)^2 - (x - 2)^2 - y^2 = 12$

$4x - 8y + 12 = 12$

$x = 2y.$

Then substitute:

$(2y)^2 + (y - 4)^2 = 16$

$4y^2 + y^2 - 8y + 16 = 16$

$5y^2 - 8y = 0$

$y(5y - 8) = 0.$

Thus $y = 0$ and $y = \frac{8}{5}$ making $x = 0$ and $x = \frac{16}{5}$.

The first value of $0$ is obviously referring to the x-coordinate of the point where the circles intersect at the origin, $D$, so the second value must be referring to the x coordinate of $P$. Since $\overline{AD}$ is the y-axis, the distance to it from $P$ is the same as the x-value of the coordinate of $P$, so the distance from $P$ to $\overline{AD}$ is $\frac{16}{5} \Rightarrow B$

Solution 2

$APMD$ obviously forms a kite. Let the intersection of the diagonals be $E$. $AE+EM=AM=2\sqrt{5}$ Let $AE=x$. Then, $EM=2\sqrt{5}-x$.


By Pythagorean Theorem, $DE^2=4^2-AE^2=2^2-EM^2$. Thus, $16-x^2=4-(2\sqrt{5}-x)^2$. Simplifying, $x=\frac{8}{\sqrt{5}}$. By Pythagoras again, $DE=\frac{4}{\sqrt{5}}$. Then, the area of $ADP$ is $DE\cdot AE=\frac{32}{5}$.


Using $4$ instead as the base, we can drop a altitude from P. $\frac{32}{5}=\frac{bh}{2}$. $\frac{32}{5}=\frac{4h}{2}$. Thus, the horizontal distance is $\frac{16}{5} \implies \boxed{\textbf{(E)}\frac{16}{5}}$


~BJHHar

Solution 3

Note that $P$ is merely a reflection of $D$ over $AM$. Call the intersection of $AM$ and $DP$ $X$. Drop perpendiculars from $X$ and $P$ to $AD$, and denote their respective points of intersection by $J$ and $K$. We then have $\triangle DXJ\sim\triangle DPK$, with a scale factor of 2. Thus, we can find $XJ$ and double it to get our answer. With some analytical geometry, we find that $XJ=\frac{8}{5}$, implying that $PK=\frac{16}{5}$.

Solution 4

As in Solution 2, draw in $DP$ and $AM$ and denote their intersection point $X$. Next, drop a perpendicular from $P$ to $AD$ and denote the foot as $Z$. $AP \cong AD$ as they are both radii and similarly $DM \cong MP$ so $APMD$ is a kite and $DX \perp XM$ by a well-known theorem.

Pythagorean theorem gives us $AM=2 \sqrt{5}$. Clearly $\triangle XMD \sim \triangle XDA \sim \triangle DMA \sim \triangle ZDP$ by angle-angle and $\triangle XMD \cong \triangle XMP$ by Hypotenuse Leg. Manipulating similar triangles gives us $PZ=\frac{16}{5}$

Solution 5

Using the double-angle formula for sine, what we need to find is $AP\cdot \sin(DAP) = AP\cdot 2\sin( DAM) \cos(DAM) = 4\cdot 2\cdot \frac{2}{\sqrt{20}}\cdot\frac{4}{\sqrt{20}} = \frac{16}{5}$.

Solution 6(LoC)

We use the Law of Cosines:

$32-32 \cos \theta = 8 + 8 \cos \theta$

$\frac{3}{5} = \cos \theta$

$2 + 2*\frac{3}{5} = \frac{16}{5}$

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

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