# Difference between revisions of "2016 AMC 8 Problems/Problem 22"

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==Solution== | ==Solution== | ||

− | The area of trapezoid <math>CBFE</math> is <math>\frac{1+3}2\cdot 4=8</math>. Next, we find the height of each triangle to calculate their area. The triangles are similar, and are in a <math>3:1</math> ratio[SOMEBODY PROVE THIS], so the height of the larger one is <math>3,</math> while the height of the smaller one is <math>1.</math> Thus, their areas are <math>\frac12</math> and <math>\frac92</math>. Subtracting these areas from the trapezoid, we get <math>8-\frac12-\frac92 =\boxed3</math>. Therefore, the answer to this problem is <math>\boxed{\textbf{(C) }3}</math> | + | The area of trapezoid <math>CBFE</math> is <math>\frac{1+3}2\cdot 4=8</math>. Next, we find the height of each triangle to calculate their area. The triangles are similar, and are in a <math>3:1</math> ratio[SOMEBODY PROVE THIS], so the height of the larger one is <math>3,</math> while the height of the smaller one is <math>1.</math> Thus, their areas are <math>\frac12</math> and <math>\frac92</math>. Subtracting these areas from the trapezoid, we get <math>8-\frac12-\frac92 =\boxed3</math>. Therefore, the answer to this problem is <math>\boxed{\textbf{(C) }3}</math> |

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==Solution 2== | ==Solution 2== |

## Revision as of 20:26, 8 January 2018

Rectangle below is a rectangle with . What is the area of the "bat wings" (shaded area)?

## Solution

The area of trapezoid is . Next, we find the height of each triangle to calculate their area. The triangles are similar, and are in a ratio[SOMEBODY PROVE THIS], so the height of the larger one is while the height of the smaller one is Thus, their areas are and . Subtracting these areas from the trapezoid, we get . Therefore, the answer to this problem is

## Solution 2

Setting coordinates!

Let ,

Now, we easily discover that line has lattice coordinates at and . Hence, the slope of line

Plugging in the rest of the coordinate points, we find that line

Doing the same process to line , we find that line .

Hence, setting them equal to find the intersection point...

.

Hence, we find that the intersection point is . Call it Z.

Now, we can see that

.

Shoelace!

Using the well known Shoelace Formula(https://en.m.wikipedia.org/wiki/Shoelace_formula), we find that the area of one of those small shaded triangles is .

Now because there are two of them, we multiple that area by to get