2021 Fall AMC 12B Problems/Problem 2

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The following problem is from both the 2021 Fall AMC 10B #2 and 2021 Fall AMC 12B #2, so both problems redirect to this page.

Problem

What is the area of the shaded figure shown below? [asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8);  pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.35,0), black+1.5, EndArrow(10)); draw(O--Y+(0,0.35), black+1.5, EndArrow(10)); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]

$\textbf{(A)}\: 4\qquad\textbf{(B)} \: 6\qquad\textbf{(C)} \: 8\qquad\textbf{(D)} \: 10\qquad\textbf{(E)} \: 12$

Solution 1 (Area Addition)

The line of symmetry divides the shaded figure into two congruent triangles, each with base $3$ and height $2.$

Therefore, the area of the shaded figure is \[2\cdot\left(\frac12\cdot3\cdot2\right)=\boxed{\textbf{(B)} \: 6}.\] ~MRENTHUSIASM ~Wilhelm Z

Solution 2 (Area Subtraction)

The area is \begin{align*} \frac{1}{2} \left( 5 - 1 \right) 5 - \frac{1}{2} \left( 5 - 1 \right) 2 & = 6 . \end{align*}

Therefore, the answer is $\boxed{\textbf{(B)} \: 6}$.

~Steven Chen (www.professorchenedu.com)

Solution 3

We start by finding the points. The outlined shape is made up of $(1,0),(3,5),(5,0),(3,2)$. By the Shoelace Theorem, we find the area to be $6$, or $\boxed{\textbf{(B)} \: 6}$.

https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem

~Taco12

~I-AM-DA-KING for the link

Solution 4

We can use Pick's Theorem. We have $4$ interior points and $6$ boundary points. By Pick's Theorem, we get $4+\frac{6}{2}-1 = 4+3-1 = 6.$ Checking our answer choices, we find our answer to be $\boxed{\textbf{(B)} \: 6}$.

~danprathab

See Also

2021 Fall AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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
All AMC 10 Problems and Solutions
2021 Fall AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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
All AMC 12 Problems and Solutions

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