# Difference between revisions of "2016 AMC 10A Problems/Problem 11"

## Problem

What is the area of the shaded region of the given $8 \times 5$ rectangle?

$[asy] size(6cm); defaultpen(fontsize(9pt)); draw((0,0)--(8,0)--(8,5)--(0,5)--cycle); filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8)); label("1",(1/2,5),dir(90)); label("7",(9/2,5),dir(90)); label("1",(8,1/2),dir(0)); label("4",(8,3),dir(0)); label("1",(15/2,0),dir(270)); label("7",(7/2,0),dir(270)); label("1",(0,9/2),dir(180)); label("4",(0,2),dir(180)); [/asy]$

$\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8$

## Solution 1

First, split the rectangle into $4$ triangles: $[asy] size(6cm); defaultpen(fontsize(9pt)); draw((0,0)--(8,0)--(8,5)--(0,5)--cycle); filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8)); label("1",(1/2,5),dir(90)); label("7",(9/2,5),dir(90)); label("1",(8,1/2),dir(0)); label("4",(8,3),dir(0)); label("1",(15/2,0),dir(270)); label("7",(7/2,0),dir(270)); label("1",(0,9/2),dir(180)); label("4",(0,2),dir(180)); draw((0,5)--(8,0)); [/asy]$

The bases of these triangles are all $1$, and their heights are $4$, $\frac{5}{2}$, $4$, and $\frac{5}{2}$. Thus, their areas are $2$, $\frac{5}{4}$, $2$, and $\frac{5}{4}$, which add to the area of the shaded region, which is $\boxed{6\frac{1}{2}}$.

## Solution 2

Find the area of the unshaded area by calculating the area of the triangles and rectangles outside of the shaded region. This will result in $40$ - $33$ $\frac{1}{2}$ = $\boxed{6$ (Error compiling LaTeX. ! File ended while scanning use of \boxed.)\frac{1}{2}}\$