Answer to Question #217358 in Geometry for Dominik

Question

Let's consider a circle in the plane with radius r and on it a rope AB at a distance r/2 from the center. Let C be a point of the minor arc AB. So the angle ACB...

A. measures 60^0

B. measures 120^0

C. has a measure that depends on the position of C

Answer 1

Let $SO$ be the distance from the line segment $AB$ to the center of the circle $O$, then $SO \perp AB$, hence, by the Pythagorean theorem, $AB=2AS=\sqrt{AO^2-SO^2}=\sqrt{r^2-\frac{1}{4}r^2}=r\sqrt{3}$.

By the cosine theorem

$⌒ACB=\angle{AOB}=\arccos{\bigg(\dfrac{AO^2+BO^2-AB^2}{2×AO×BO}\bigg)}=\arccos{\bigg(\dfrac{-r^2}{2r^2}\bigg)}=\arccos{(-0.5)}=120\degree$

Answer is B.