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 21:26, 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