2020 AMC 12A Problems/Problem 14

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 1

$ACEG$ is a square. WLOG $AB = 1,$ then using Law of Cosines, $AC^2 = [ACEG] = 1^2 + 1^2 - 2 \cos{135} = 2 + \sqrt{2}.$ The area of the octagon is just $[ACEG]$ plus the area of the four congruent (by symmetry) isosceles triangles, all an angle of $135$ in between two sides of length 1. Now, $\dfrac{m}{n} = \dfrac{2 + \sqrt{2}}{2 + \sqrt{2} + 4 \cdot \tfrac{1}{2} \sin{135}} = \dfrac{2 + \sqrt{2}}{2 + 2 \sqrt{2}} = \dfrac{\sqrt{2}}{2}.$ The answer is $\boxed{\textbf{(B)}}.$

Solution 2

Refer to the diagram. Call one of the side lengths of the square $s$ . Since quadrilateral ACEG is a square, the area of the square would just be $s^2$ , which we can find by applying Law of Cosines on one of the four triangles. Assume each of the sides of the octagon has length $1$ . Since each angle measures $135^{\circ}$ in an octagon, then $s^2 = 1^2+1^2-2*\cos(135^{\circ}) = 2+\sqrt{2}$

There are many ways to find the area of the octagon, but one way is to split the octagon into two trapezoids and one rectangle. We can easily compute AF to be $1+\sqrt{2}$ from splitting one of the sides into two $45-45-90$ triangles. So the area of the octagon is $2*\frac{1+\sqrt{2}}{2}+1+\sqrt{2} \Rightarrow 2+2\sqrt{2}$ .

Hence, $\frac{m}{n} = \frac{2+\sqrt{2}}{2+2\sqrt{2}}$ $\Rightarrow \frac{2+\sqrt{2}}{2+2\sqrt{2}} \cdot \frac{2-2\sqrt{2}}{2-2\sqrt{2}}$ $\Rightarrow \frac{-2\sqrt{2}}{-4}$ $\Rightarrow \boxed{\frac{\sqrt{2}}{2}}$

~Solution by IronicNinja

Solution 3 (Deriving Formulas)

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

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.

Solution 3 (Elementary Geometry)

WLOG, let $AE=1$ . Let the intersection of $AF$ and $GD$ be point $I$ . $GIF$ is an isosceles right triangle ( $m\angle HGF=135^{\circ}$ ), so $IG=IF=\frac{1}{\sqrt{2}}$ . The distance between each side and the center is then $IF+\frac{1}{2}GH=\frac{1}{2}+\frac{1}{\sqrt{2}}$ . $ABCDEFGH$ is 8 triangles of base 1 and altitude $\frac{1}{2}+\frac{1}{\sqrt{2}}$ , so its area is $8(\frac{1}{2})(\frac{1}{2}+\frac{1}{\sqrt{2}})$ or $2+2\sqrt{2}$ Similarly, $ACEG$ is clearly a square of area $GA^2$ . By the Pythagorean Theorem, $GA^2=GI^2+IA^2=(\frac{1}{\sqrt{2}})^2+(1+\frac{1}{\sqrt{2}})^2$ or $2+\sqrt{2}$ $\frac{m}{n} = \frac{2+\sqrt{2}}{2+2\sqrt{2}}$ From some fast manipulations, see that it is $\boxed{\textbf{B)}\frac{\sqrt{2}}{{2}}}$

~~BJHHar

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