Difference between revisions of "2005 AMC 8 Problems/Problem 23"

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<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi </math>
 
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi </math>
  
==Easier and More Logical Solution==
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==Solution==
First, we notice half a square so first let's create a square. Once we have a square, we will have a full circle. This circle has a diameter of 4 which will be the side of the square. The area would be 4*4 = 16. Divide 16 by 2 to get the original shape and you get <math>\boxed{8}</math>
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First, we notice half a square so first let's create a square. Once we have a square, we will have a full circle. This circle has a diameter of 4 which will be the side of the square. The area would be <math>4\cdot 4 = 16.</math> Divide 16 by 2 to get the original shape and you get <math>\boxed{8}</math>
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==Solution 2==
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We can figure out the radius of the semicircle because the question states that the area of the semicircle is <math> 2\pi</math> and we can multiply it by 2 to get <math> 4\pi </math> which we can see it is 2 from the formula. Draw line segment OD such that it is the midsegment of triangle ABC, using the midsegment theorem we can see that line segment AC = <math>2*2=4</math>. Since triangle ABC is an isosceles right triangle we can calculate the area to be <math>\frac{4^2}{2}</math> = <math>\boxed{8}</math>
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==Video Solution==
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https://www.youtube.com/watch?v=cNbXCQXUc6E  ~David
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==Video Solution by OmegaLearn==
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https://youtu.be/j3QSD5eDpzU?t=1116
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~ pi_is_3.14
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==See Also==
 
==See Also==
 
{{AMC8 box|year=2005|num-b=22|num-a=24}}
 
{{AMC8 box|year=2005|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 09:14, 18 October 2024

Problem

Isosceles right triangle $ABC$ encloses a semicircle of area $2\pi$. The circle has its center $O$ on hypotenuse $\overline{AB}$ and is tangent to sides $\overline{AC}$ and $\overline{BC}$. What is the area of triangle $ABC$?

[asy]pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(circle(o, 2)); clip(a--b--c--cycle); draw(a--b--c--cycle); dot(o); label("$C$", c, NW); label("$A$", a, NE); label("$B$", b, SW);[/asy]

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi$

Solution

First, we notice half a square so first let's create a square. Once we have a square, we will have a full circle. This circle has a diameter of 4 which will be the side of the square. The area would be $4\cdot 4 = 16.$ Divide 16 by 2 to get the original shape and you get $\boxed{8}$

Solution 2

We can figure out the radius of the semicircle because the question states that the area of the semicircle is $2\pi$ and we can multiply it by 2 to get $4\pi$ which we can see it is 2 from the formula. Draw line segment OD such that it is the midsegment of triangle ABC, using the midsegment theorem we can see that line segment AC = $2*2=4$. Since triangle ABC is an isosceles right triangle we can calculate the area to be $\frac{4^2}{2}$ = $\boxed{8}$

Video Solution

https://www.youtube.com/watch?v=cNbXCQXUc6E ~David

Video Solution by OmegaLearn

https://youtu.be/j3QSD5eDpzU?t=1116

~ pi_is_3.14


See Also

2005 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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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