Four sets of pairs of parallel lines are graphed. Each line shares an x-intercept with one other line and a y-intercept with one other line.

The lines appear to show an 8-pointed star.

If the star has 8 fold rotational symmetry and all 8 points of the star are equal size, give the slope of the blue line.

The answer is 2.414213562.

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Since the star has $8$ -fold rotational symmetry, the center is a regular octagon with interior angles of $135°$ . The angle $\theta$ between the blue line and the $x$ -axis is half this angle, so $\theta = \frac{135°}{2}$ , and since $m = \tan \theta$ , the slope $m$ of the blue line is is $m = \tan \frac{135°}{2} = \sqrt{2} + 1 \approx \boxed{2.414213562}$ .