Difference between revisions of "2015 AMC 8 Problems/Problem 21"
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− | In the given figure hexagon <math>ABCDEF</math> is equiangular, <math>ABJI</math> and <math>FEHG</math> are squares with areas <math>18</math> and <math>32</math> respectively, <math>\triangle JBK</math> is equilateral and <math>FE=BC</math>. What is the area of <math>\triangle KBC</math>? | + | ==Problem== |
+ | |||
+ | In the given figure, hexagon <math>ABCDEF</math> is equiangular, <math>ABJI</math> and <math>FEHG</math> are squares with areas <math>18</math> and <math>32</math> respectively, <math>\triangle JBK</math> is equilateral and <math>FE=BC</math>. What is the area of <math>\triangle KBC</math>? | ||
− | |||
<asy> | <asy> | ||
draw((-4,6*sqrt(2))--(4,6*sqrt(2))); | draw((-4,6*sqrt(2))--(4,6*sqrt(2))); | ||
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</asy> | </asy> | ||
− | == Solution == | + | <math>\textbf{(A) }6\sqrt{2}\quad\textbf{(B) }9\quad\textbf{(C) }12\quad\textbf{(D) }9\sqrt{2}\quad\textbf{(E) }32</math>. |
+ | |||
+ | == Solution 1 == | ||
+ | |||
+ | Clearly, since <math>\overline{FE}</math> is a side of a square with area <math>32</math>, <math>\overline{FE} = \sqrt{32} = 4 \sqrt{2}</math>. Now, since <math>\overline{FE} = \overline{BC}</math>, we have <math>\overline{BC} = 4 \sqrt{2}</math>. | ||
+ | |||
+ | Now, <math>\overline{JB}</math> is a side of a square with area <math>18</math>, so <math>\overline{JB} = \sqrt{18} = 3 \sqrt{2}</math>. Since <math>\Delta JBK</math> is equilateral, <math>\overline{BK} = 3 \sqrt{2}</math>. | ||
+ | |||
+ | Lastly, <math>\Delta KBC</math> is a right triangle. We see that <math>\angle JBA + \angle ABC + \angle CBK + \angle KBJ = 360^{\circ} \rightarrow 90^{\circ} + 120^{\circ} + \angle CBK + 60^{\circ} = 360^{\circ} \rightarrow \angle CBK = 90^{\circ}</math>, so <math>\Delta KBC</math> is a right triangle with legs <math>3 \sqrt{2}</math> and <math>4 \sqrt{2}</math>. Now, its area is <math>\dfrac{3 \sqrt{2} \cdot 4 \sqrt{2}}{2} = \dfrac{24}{2} = \boxed{\textbf{(C)}~12}</math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | Since <math>\overline{FE} = \sqrt{32}</math>, and <math>\overline{FE} = \overline{BC}</math>, <math>\overline{BC} = 4\sqrt{2}</math>. Meanwhile, <math>\overline{JB} = 3\sqrt{2}</math>, and since <math>\triangle JBK</math> is equilateral, <math>\overline{BK} = 3\sqrt{2}</math>. If <math>ABCDEF</math> is equiangular, <math>\angle ABC = \frac{180 \cdot (n-2)}{n} = 120^{\circ}</math>, where <math>n</math> is the number of sides of the shape. Adding all the angles around <math>B</math> gives <math>270^{\circ}</math>, so <math>\angle KBC = 360 - 90 = 270^{\circ}</math>. Because <math>\triangle KBC</math> is right, the area of <math>\triangle KBC = \frac{4\sqrt{2} \cdot 3\sqrt{2}} {2} = 12</math>. Therefore, the answer is <math>\boxed{\textbf{(C)}~12}</math>. ~strongstephen | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/mUBhWTlvuLQ | ||
− | + | ~savannahsolver | |
− | + | ==See Also== | |
− | + | {{AMC8 box|year=2015|num-b=20|num-a=22}} | |
+ | {{MAA Notice}} |
Latest revision as of 19:36, 17 January 2024
Problem
In the given figure, hexagon is equiangular, and are squares with areas and respectively, is equilateral and . What is the area of ?
.
Solution 1
Clearly, since is a side of a square with area , . Now, since , we have .
Now, is a side of a square with area , so . Since is equilateral, .
Lastly, is a right triangle. We see that , so is a right triangle with legs and . Now, its area is .
Solution 2
Since , and , . Meanwhile, , and since is equilateral, . If is equiangular, , where is the number of sides of the shape. Adding all the angles around gives , so . Because is right, the area of . Therefore, the answer is . ~strongstephen
Video Solution
~savannahsolver
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 AJHSME/AMC 8 Problems and Solutions |
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