To find the side of a square whose two vertices lie on a circle

Question

Two vertices of a square lie on a circle of radius r,and two other vertices lie on a tangent to this circle. Then each side of a square is - $$(A)\frac{3r}{2}\quad (B)\frac{4r}{3}\quad (C)\frac{6r}{5}\quad (D)\frac{8r}{5}$$ Source

After drawing the diagram I found that two vertices lie outside the circle and two on the circle ,but I can't tell whether the diagonal of the square passes through the center of the circle. I am stuck here.

If the diagonal(s) of the square passed through the center of the circle, then each side of the square would be $\sqrt2r$, which is not an option. — Barry Cipra, Apr 25 '18 at 16:06
If you can use trigonometry solve $r+r\cos \alpha=2 r \sin \alpha$ — N74, Apr 25 '18 at 16:18
Possible duplicate of Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers — Steven Stadnicki, Apr 25 '18 at 18:01
(Essentially a duplicate, since the geometry in the linked problem is the same and the answer to this question is at the core of the answer to that one) — Steven Stadnicki, Apr 25 '18 at 18:02

score 2 · Accepted Answer · answered Apr 25 '18 at 16:15

2

If you draw a diameter for the circle from the point of tangency, it bisects the chord connecting the two points of the square that lie on the circle. The chord theorem says

$$\left(s\over2\right)^2=s(2r-s)$$

from which you can find that

$$s={8r\over5}$$

answered Apr 25 '18 at 16:15

Barry Cipra

81,321

Nice, I just looked up this theorem, I am having a hard time without this theorem. – jonsno Apr 25 '18 at 16:18
https://www.dummies.com/education/math/geometry/how-to-use-the-chord-chord-power-theorem/ – Clemens Bartholdy Jul 05 '21 at 11:50

To find the side of a square whose two vertices lie on a circle

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