Octagon inside a square!

I want to know a simple method for the following question: There is a square of side 1 unit.Mark the mid points of the sides,say X,Y,Z,W of sides AB,BC,CD,AD respectively.Now form a triangle with vertices X,C, and D.Similarly join Y,A and D forming a triangle.Do the same for other two.Now compute the area of the octagon formed at the center of the square.Hope u understand the problem!?

#Geometry #MathProblem #Math

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Comments

Use coordinates. To avoid fractions, I'll scale everything by a factor of $12$ .

WLOG, let $A(0,0), B(12,0), C(12,12), D(0,12)$ . Then, $X(6,0), Y(12,6), X(6,12), W(0,6)$ .

Let $V_1, \ldots , V_8$ be the vertices of the octagon going counterclockwise, with $V_1$ being closest to $W$ .

Note that there is $8$ -fold symmetry about the center $M(6,6)$ .

Line $AZ$ is $y = 2x$ . Line $DX$ is $y = 12-2x$ . Their intersection is $V_1(3,6)$ .

By symmetry about $M(6,6)$ , the other odd numbered vertices are $V_3(6,3), V_5(9,6), V_7(6,9)$ .

Line $BW$ is $y = -\tfrac{1}{2}x+6$ . Line $DX$ is $y = 12-2x$ . Their intersection is $V_2(4,4)$ .

By symmetry about $M(6,6)$ , the other even numbered vertices are $V_4(8,4), V_6(8,8), V_8(4,8)$ .

From here, it is easy to calculate the area of one "slice" of the scaled octagon: $[\Delta MV_1V_2] = \tfrac{1}{2} \cdot 3 \cdot 2 = 3$ . Therefore, the scaled octagon has area $8 \cdot 3 = 24$ . Hence, the original octagon has area $\dfrac{24}{12^2} = \boxed{\dfrac{1}{6}}$ .

Jimmy Kariznov - 7 years, 10 months ago

Coincidentally, this was a problem from the 1989 Dutch finals, it was put in a collection of the 100 best problems from Dutch olympiads over the years. I tried to avoid coordinates, as it's clear that that's going to work, but I tried to find a more elegant solution, e.g. by determining the area of the part of the square that's not part of the octagon. I didn't manage to do that though. If I encountered such a problem on an olympiad, I'd probably go for the 'safe' coordinate method.

Tim Vermeulen - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$