2020 AMC 12A Problem 2

Question

The acronym AMC is shown in the rectangular grid below with grid lines spaced

1

unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?

17

15 + 2\sqrt{2}

13 + 4\sqrt{2}

11 + 6\sqrt{2}

21

A segment spanning one unit right and one unit up has length

\sqrt{2}

Total the horizontal and vertical pieces separately from the diagonal pieces

Split each letter into its segments. The

A

is a diagonal of length

2\sqrt{2},

a vertical of length

2,

and a crossbar of length

1.

The

M

has two verticals of length

2

and two diagonals of length

\sqrt{2}

each. The

C

is three sides of length

2.

The straight pieces total

2 + 1 + 2 + 2 + 2 + 2 + 2 = 13,

and the diagonal pieces total

2\sqrt2 + \sqrt2 + \sqrt2 = 4\sqrt2.

The sum is

13 + 4\sqrt2.

Thus, C is the correct answer.

Problem 2 in other years