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Difference between revisions of "2020 AMC 12A Problems/Problem 14"

(Created page with "Problem 14 Regular octagon <math>ABCDEFGH</math> has area <math>n</math>. Let <math>m</math> be the area of quadrilateral <math>ACEG</math>. What is <math>\tfrac{m}{n}?</math>...")
 
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<math>\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}</math>
 
<math>\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}</math>
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== Solution (Formulaic) ==
 +
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<asy>
 +
draw((1, 2.41421356)--(2.41421356, 1)--(2.41421356, -1)--(1, -2.41421356)--(-1, -2.41421356)--(-2.41421356, -1)--(-2.41421356, 1)--(-1, 2.41421356)--cycle);
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draw((2.41421356, 1)--(1, -2.41421356)--(-2.41421356, -1)--(-1, 2.41421356)--cycle);
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label("A",(-1, 2.41421356),NW);
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label("B",(1, 2.41421356),NE);
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label("C",(2.41421356, 1),NE);
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label("D",(2.41421356, -1),SE);
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label("E",(1,-2.41421356),SE);
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label("F",(-1,-2.41421356),SW);
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label("G",(-2.41421356,-1),SW);
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label("H",(-2.41421356,1),NW);
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</asy>
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 +
The first thing to notice is that <math>ACEG</math> is a square. This is because, as <math>\bigtriangleup ABC \cong \bigtriangleup CDE \cong \bigtriangleup EFG \cong \bigtriangleup GHE</math>, they all have the same base, meaning that <math>AC = DE = EG = GA</math>. Hence, we have that it is a square. To determine the area of this square, we can determine the length of its diagonals.
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In order to do this, we first determine the area of the octagon. Letting the side length be <math>a</math>, we can create a square of length <math>s</math> around it (see figure).
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<asy>
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draw((1, 2.41421356)--(2.41421356, 1)--(2.41421356, -1)--(1, -2.41421356)--(-1, -2.41421356)--(-2.41421356, -1)--(-2.41421356, 1)--(-1, 2.41421356)--cycle);
 +
label("a",(1, 2.41421356)--(2.41421356, 1),S);
 +
draw((-2.41421356, 2.41421356)--(2.41421356, 2.41421356)--(2.41421356, -2.41421356)--(-2.41421356, -2.41421356)--cycle);
 +
label("s",(-2.41421356, 2.41421356)--(2.41421356, 2.41421356),N);
 +
</asy>
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 +
Creating a small square of side length <math>a</math> from the corners of this figure gives us an area of <math>a^2</math>. Thus, <math>s^2 - a^2 = n</math> where <math>n</math> is the area of the octagon. We know from the Pythagorean Theorem that <math>s = a + \frac{a}{\sqrt{2}}</math>, meaning that <math>n = (a + \frac{a}{\sqrt{2}})^2 - a^2 = 2a^2(1+\sqrt{2})</math>.
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Dividing this by <math>8</math> gives us the area of each triangular segment which makes up the octagon. Further dividing by <math>2</math> gives us the area of a smaller segment consisting of the right triangle with legs of the apothem and <math>\frac{a}{2}</math>. Using the area of a triangle as <math>\frac{1}{2}bh</math>, we can determine the length of apothem <math>r</math> from
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<cmath>\frac{2a^2(1+\sqrt{2})}{8 \times 2} = \frac{\frac{a}{2}\times r}{2}</cmath>
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<cmath>\frac{4a^2(1+\sqrt{2})}{8 } = ar</cmath>
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<cmath>\frac{a(1+\sqrt{2}}{2} = r</cmath>.
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From the apothem, we can once again use the Pythagorean Theorem, giving us the length of the circumradius <math>R</math>.
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<cmath>R^2 = (\frac{a(1+\sqrt{2})}{2})^2 + (\frac{a}{2})^2</cmath>
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<cmath>R^2 = \frac{a^2(1+\sqrt{2})^2}{4} + \frac{a^2}{4}</cmath>
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<cmath>R^2 = \frac{a^2(3+2\sqrt{2})}{4} + \frac{a^2}{4}</cmath>
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<cmath>R^2 = \frac{a^2(4+2\sqrt{2})}{4}</cmath>
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<cmath>R = \sqrt{\frac{a^2(4+2\sqrt{2})}{4}} = \frac{a\sqrt{4 + 2\sqrt2}}{2}</cmath>.
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Doubling this gives us the diagonal of both the square and the octagon. From here, we can use the formula <math>A = \frac{1}{2}d^2</math> for the area of the square:
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<cmath>A = \frac{(a\sqrt{4 + 2\sqrt2})^2}{2}</cmath>
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<cmath>A = \frac{a^2(4+2\sqrt{2})}{2} = m</cmath>.
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Thus we now only need to find the ratio <math>\frac{m}{n}</math>. This can be easily done through some algebra:
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<cmath>\frac{m}{n} = \frac{a^2(4+2\sqrt{2})}{2(2a^2(1+\sqrt{2})}</cmath>
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<cmath>\frac{m}{n} = \frac{(4+2\sqrt{2})}{4+4\sqrt{2}}</cmath>
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<cmath>\frac{m}{n} = \frac{(2+\sqrt{2})}{2+2\sqrt{2}}</cmath>
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Rationalizing the denominator by multiplying by the conjugate, we get <math>\frac{m}{n} = \boxed{\textbf{(B) } \frac{\sqrt{2}}{2}}</math>. ~ciceronii
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*<math>\textbf{Note:}</math> this can more easily be done if you know any of these formulas. This was an entire derivation of the area of an octagon, it's apothem, and it's circumradius, but it can be much simpler if you have any of these memorized.
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==See Also==
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{{AMC12 box|year=2020|ab=A|num-b=13|num-a=15}}
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{{MAA Notice}}

Revision as of 16:20, 1 February 2020

Problem 14 Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$

$\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}$

Solution (Formulaic)

[asy] draw((1, 2.41421356)--(2.41421356, 1)--(2.41421356, -1)--(1, -2.41421356)--(-1, -2.41421356)--(-2.41421356, -1)--(-2.41421356, 1)--(-1, 2.41421356)--cycle); draw((2.41421356, 1)--(1, -2.41421356)--(-2.41421356, -1)--(-1, 2.41421356)--cycle); label("A",(-1, 2.41421356),NW); label("B",(1, 2.41421356),NE); label("C",(2.41421356, 1),NE); label("D",(2.41421356, -1),SE); label("E",(1,-2.41421356),SE); label("F",(-1,-2.41421356),SW); label("G",(-2.41421356,-1),SW); label("H",(-2.41421356,1),NW); [/asy]

The first thing to notice is that $ACEG$ is a square. This is because, as $\bigtriangleup ABC \cong \bigtriangleup CDE \cong \bigtriangleup EFG \cong \bigtriangleup GHE$, they all have the same base, meaning that $AC = DE = EG = GA$. Hence, we have that it is a square. To determine the area of this square, we can determine the length of its diagonals.

In order to do this, we first determine the area of the octagon. Letting the side length be $a$, we can create a square of length $s$ around it (see figure). [asy] draw((1, 2.41421356)--(2.41421356, 1)--(2.41421356, -1)--(1, -2.41421356)--(-1, -2.41421356)--(-2.41421356, -1)--(-2.41421356, 1)--(-1, 2.41421356)--cycle); label("a",(1, 2.41421356)--(2.41421356, 1),S); draw((-2.41421356, 2.41421356)--(2.41421356, 2.41421356)--(2.41421356, -2.41421356)--(-2.41421356, -2.41421356)--cycle); label("s",(-2.41421356, 2.41421356)--(2.41421356, 2.41421356),N); [/asy]

Creating a small square of side length $a$ from the corners of this figure gives us an area of $a^2$. Thus, $s^2 - a^2 = n$ where $n$ is the area of the octagon. We know from the Pythagorean Theorem that $s = a + \frac{a}{\sqrt{2}}$, meaning that $n = (a + \frac{a}{\sqrt{2}})^2 - a^2 = 2a^2(1+\sqrt{2})$.

Dividing this by $8$ gives us the area of each triangular segment which makes up the octagon. Further dividing by $2$ gives us the area of a smaller segment consisting of the right triangle with legs of the apothem and $\frac{a}{2}$. Using the area of a triangle as $\frac{1}{2}bh$, we can determine the length of apothem $r$ from

\[\frac{2a^2(1+\sqrt{2})}{8 \times 2} = \frac{\frac{a}{2}\times r}{2}\] \[\frac{4a^2(1+\sqrt{2})}{8 } = ar\] \[\frac{a(1+\sqrt{2}}{2} = r\].

From the apothem, we can once again use the Pythagorean Theorem, giving us the length of the circumradius $R$.

\[R^2 = (\frac{a(1+\sqrt{2})}{2})^2 + (\frac{a}{2})^2\] \[R^2 = \frac{a^2(1+\sqrt{2})^2}{4} + \frac{a^2}{4}\] \[R^2 = \frac{a^2(3+2\sqrt{2})}{4} + \frac{a^2}{4}\] \[R^2 = \frac{a^2(4+2\sqrt{2})}{4}\] \[R = \sqrt{\frac{a^2(4+2\sqrt{2})}{4}} = \frac{a\sqrt{4 + 2\sqrt2}}{2}\].

Doubling this gives us the diagonal of both the square and the octagon. From here, we can use the formula $A = \frac{1}{2}d^2$ for the area of the square:

\[A = \frac{(a\sqrt{4 + 2\sqrt2})^2}{2}\] \[A = \frac{a^2(4+2\sqrt{2})}{2} = m\].

Thus we now only need to find the ratio $\frac{m}{n}$. This can be easily done through some algebra:

\[\frac{m}{n} = \frac{a^2(4+2\sqrt{2})}{2(2a^2(1+\sqrt{2})}\] \[\frac{m}{n} = \frac{(4+2\sqrt{2})}{4+4\sqrt{2}}\] \[\frac{m}{n} = \frac{(2+\sqrt{2})}{2+2\sqrt{2}}\]

Rationalizing the denominator by multiplying by the conjugate, we get $\frac{m}{n} = \boxed{\textbf{(B) } \frac{\sqrt{2}}{2}}$. ~ciceronii

  • $\textbf{Note:}$ this can more easily be done if you know any of these formulas. This was an entire derivation of the area of an octagon, it's apothem, and it's circumradius, but it can be much simpler if you have any of these memorized.

See Also

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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 AMC 12 Problems and Solutions

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