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

Revision as of 12:02, 14 February 2014

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 $\overline{AD}$ ?

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

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

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 3

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 $PZ=\frac{16}{5}$

@@ Line 14: / Line 14: @@
 <math>(x-2)^2 + y^2 = 2^2</math>
+Subtract the second equation from the first:
+<math>x^2 + (y - 4)^2 - (x - 2)^2 - y^2 = 12</math>
-Algebraically manipulating the second equation gives:
+<math>4x - 8y + 12 = 12</math>
-<math>y^2 = 2^2 - (x-2)^2</math>
+<math>x = 2y.</math>
-<math>y^2 = (2-(x-2)(2+(x-2))</math>
+Then substitute:
+<math>(2y)^2 + (y - 4)^2 = 16</math>
-<math>y^2 = (4-x)(x)</math>
+<math>4y^2 + y^2 - 8y + 16 = 16</math>
-<math>y = \sqrt{4x - x^2}</math>
+<math>5y^2 - 8y = 0</math>
+<math>y(5y - 8) = 0.</math>
-Substituting this back into the first equation:
+Thus <math>y = 0</math> and <math>y = \frac{8}{5}</math> making <math>x = 0</math> and <math>x = \frac{16}{5}</math>.
-<math>x^2 + (\sqrt{4x - x^2} - 4)^2 = 4^2</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>
-<math>x^2 + 4x - x^2 - 8\sqrt{4x - x^2} + 16 = 16</math>
-<math>4x - 8\sqrt{4x - x^2} = 0</math>
-<math>4x = 8\sqrt{4x - x^2}</math>
-<math>16x^2 = 64(4x - x^2)</math>
-<math>16x^2 = 256x - 64x^2</math>
-<math>80x^2 - 256x = 0</math>
-<math>x(80x - 256) = 0</math>
-Solving each factor for 0 yields <math>x = 0 , \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>
 ==Solution 2==

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