2021 AMC 10B Problems/Problem 23

Problem

A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$ , where $a$ and $b$ are positive integers. What is $a+b$ ? $[asy] /* Made by samrocksnature */ draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); fill((2,0)--(0,2)--(0,0)--cycle, black); fill((6,0)--(8,0)--(8,2)--cycle, black); fill((8,6)--(8,8)--(6,8)--cycle, black); fill((0,6)--(2,8)--(0,8)--cycle, black); fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); filldraw(circle((2.6,3.31),0.47),gray); [/asy]$

$\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72$

Video Solution by OmegaLearn (Similar Triangles and Area Calculations)

https://youtu.be/-UvivZ0UA1U

~ pi_is_3.14

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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
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2021 AMC 10B Problems/Problem 23

Problem

Video Solution by OmegaLearn (Similar Triangles and Area Calculations)