Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (04)

Problem

As shown in the figure, if there is a round field with a diameter of $100 \ m$ and it is divided equally among $5$ people, find the length of each of the red and blue line segments.

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$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$

$$ $$

Solution

If the diameter of the round field is $d$, its area $S$ is

$$S=\frac{πd^2}{4}.$$

Divide this into $5$ equal parts. Then, the area per person $s$ is

$$s=\frac{πd^2}{4×5}.$$

Therefore, the length $l$ of one side (red line segment) of the square field in the middle is

$$l=\sqrt{\frac{πd^2}{20}}. \qquad (*)$$

Substituting $d=100$ and $π=3.14$, we have

$$l=\sqrt{\frac{3.16×10000}{20}}=\sqrt{1570}≒ 39.6232 …$$

Next, draw a right triangle as shown.

Letting the length of the blue line segment be $x$, then from the Pythagorean theorem,

$$(x-\frac{l}{2})^2+(\frac{l}{2})^2=(\frac{d}{2})^2,$$

$$x^2-lx+\frac{2l^2-d^2}{4}=0,$$

$$∴ \ x=\frac{\sqrt{d^2-l^2 }+l}{2}. \qquad (∵ \ x>0) \qquad (**)$$

Substituting $d=100$ and $l=\sqrt{1570}$ into $(**)$, we have

$$x=\frac{\sqrt{10000-1570}+\sqrt{1570}}{2}≒65.7191…$$

$$(Answer) \quad Blue:\ approximately \ 65.72 \ m, \qquad Red: \ approximately \ 39.62 \ m.$$

Reference

Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.38; pp.359-360.