Difference between revisions of "2012 AMC 12A Problems/Problem 12"

Revision as of 19:40, 3 January 2020

Problem

A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$ . Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$ . What is the side length of this square?

$\textbf{(A)}\ \frac{\sqrt{10}+5}{10}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{5}\qquad\textbf{(C)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}\qquad\textbf{(E)}\ \frac{9-\sqrt{17}}{5}$

Solution 1

$[asy] unitsize(15mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real a=1; real b=2; pair O=(0,0); pair A=(-(sqrt(19)-2)/5,1); pair B=((sqrt(19)-2)/5,1); pair C=((sqrt(19)-2)/5,1+2(sqrt(19)-2)/5); pair D=(-(sqrt(19)-2)/5,1+2(sqrt(19)-2)/5); pair E=(-(sqrt(19)-2)/5,0); path inner=Circle(O,a); path outer=Circle(O,b); draw(outer); draw(inner); draw(A--B--C--D--cycle); draw(O--D--E--cycle); label("$A$",D,NW); label("$E$",E,SW); label("$O$",O,SE); label("$s+1$",(D--E),W); label("$\frac{s}{2}$",(E--O),S); pair[] ps={A,B,C,D,E,O}; dot(ps); [/asy]$

The circles have radii of $1$ and $2$ . Draw the triangle shown in the figure above and write expressions in terms of $s$ (length of the side of the square) for the sides of the triangle. Because $AO$ is the radius of the larger circle, which is equal to $2$ , we can write the Pythagorean Theorem.

$\begin{align*} \left( \frac{s}{2} \right) ^2 + (s+1)^2 &= 2^2\\ \frac14 s^2 + s^2 + 2s + 1 &= 4\\ \frac54 s^2 +2s - 3 &= 0\\ 5s^2 + 8s - 12 &=0 \end{align*}$

Use the quadratic formula.

$s = \frac{-8+\sqrt{8^2-4(5)(-12)}}{10} = \frac{-8+\sqrt{304}}{10} = \frac{-8+4\sqrt{19}}{10} = \boxed{\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}}$

Solution 2

Using the diagram above, we look at the top-right vertex of the square. Let us call this point $(x,y)$ . Then, we that since the square is symmetrical over the y-axis, that the y value is equal to $2x+1$ , since we can multiply the x value(which is half of $s$ ) by two to get $s$ , and we add one since the square lies one unit above the origin. Now, all we must do is find the intersection of the larger circle, $x^2 + y^2 = 4$ , and the line $y=2x+1$ . Substituting the second equation into the first, we get:

$5x^2 +4x -3 = 0$

Using the quadratic formula, we arrive with $x=\frac{-4 \pm 2\sqrt{19}}{10}$ . However, recall that the x value is only one half of the side length. Multiplying this value by $2$ , then, and using only the positive root(since the top right vertex of the square has a positive x value), we get:

$\frac{-4 + 2\sqrt{19}}{5} \Rightarrow \boxed{\textbf{(D)}}$

Solution 3

Like the previous solutions, we can say that the square is symmetrical with respect to the y-axis.

Since CD passes through point (0,1), let C be (-x,1) and D be (x,1)

Point B is (-x, $\sqrt(4-x^2)$ ) and Point A is (x, $\sqrt{(4-x^2)}$ ). Now, we see that AB = 2x and BC = $\sqrt{(4-x^2)}$ -1

Setting those two equal, we get the same equation as the previous solutions: $5x^2 +4x -3 = 0$ and then solve the same way.

-Conantwiz2023

@@ Line 63: / Line 63: @@
   Since CD passes through point (0,1), let C be (-x,1) and D be (x,1)
-Point B is (-x, <math>\sqrt(4-x^2)</math>) and Point A is (x,<math>\sqrt(4-x^2)</math>). Now, we see that AB = 2x and BC = <math>\sqrt(4-x^2)</math>-1
+Point B is (-x, <math>\sqrt(4-x^2)</math>) and Point A is (x,<math>\sqrt{(4-x^2)}</math>). Now, we see that AB = 2x and BC = <math>\sqrt{(4-x^2)}</math>-1
 Setting those two equal, we get the same equation as the previous solutions: <math>5x^2 +4x -3 = 0</math> and then solve the same way.

2012 AMC 12A (Problems • Answer Key • Resources)
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