2020 AMC 12B Problems/Problem 10

Problem 10

In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$

$\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$

Solution

Place circle $\omega$ in the Cartesian plane such that the center lies on the origin. Then we can easily find the equation for $\omega$ as $x^2+y^2=\frac{1}{4}$ , because it is not translated and the radius is $\frac{1}{2}$ .

We have $A=\left(-\frac{1}{2}, \frac{1}{2}\right)$ and $M=\left(0, -\frac{1}{2}\right)$ . The slope of the line passing through these two points is $\frac{\frac{1}{2}+\frac{1}{2}}{-\frac{1}{2}-0}=\frac{1}{-\frac{1}{2}}=-2$ , and the $y$ -intercept is simply $M$ . This gives us the line passing through both points as $y=-2x-\frac{1}{2}$ .

We substitute this into the equation for the circle to get $x^2+\left(-2x-\frac{1}{2}\right)^2=\frac{1}{4}$ , or $x^2+4x^2+2x+\frac{1}{4}=\frac{1}{4}$ . Simplifying gives $x(5x+2)=0$ . The roots of this quadratic are $x=0$ and $x=-\frac{2}{5}$ , but if $x=0$ we get point $M$ , so we only want $x=-\frac{2}{5}.

We plug this back into the linear equation to find$ (Error compiling LaTeX. Unknown error_msg)y=\frac{3}{10} $, and so$ P=\left(-\frac{2}{5}, \frac{3}{10}\right) $. Finally, we use distance formula on$ A $and$ P $to get$ AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\mathbf{(B) } \frac{\sqrt{5}}{10}}$.

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2020 AMC 12B Problems/Problem 10

Problem 10

Solution