# 2018 AMC 12B Problems/Problem 4

## Problem

A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$. What is the area of the circle? $\textbf{(A) }25\pi \qquad \textbf{(B) }50\pi \qquad \textbf{(C) }75\pi \qquad \textbf{(D) }100\pi \qquad \textbf{(E) }125\pi \qquad$

## Solution

Let $O$ be the center of the circle, $\overline{AB}$ be the chord, and $M$ be the midpoint of $\overline{AB},$ as shown below. $[asy] /* Made by MRENTHUSIASM */ size(200); pair O, A, B, M; O = (0,0); A = (-5,5); B = (5,5); M = midpoint(A--B); draw(Circle(O,5sqrt(2))); dot("O", O, 1.5*S, linewidth(4.5)); dot("A", A, 1.5*NW, linewidth(4.5)); dot("B", B, 1.5*NE, linewidth(4.5)); dot("M", M, 1.5*N, linewidth(4.5)); draw(A--B^^M--O^^A--O^^M--O^^B--O); label("5", midpoint(A--M), 1.5*N); label("5", midpoint(B--M), 1.5*N); label("5", midpoint(O--M), 1.5*E); label("r", midpoint(O--A), 1.5*SW); label("r", midpoint(O--B), 1.5*SE); [/asy]$ Note that $\overline{OM}\perp\overline{AB}.$ Since $OM=AM=BM=5,$ we conclude that $\triangle OMA$ and $\triangle OMB$ are congruent isosceles right triangles. It follows that $r=5\sqrt2,$ so the area of $\odot O$ is $\pi r^2=\boxed{\textbf{(B) }50\pi}$.

~MRENTHUSIASM

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