1. **Problem statement:** Points A and B lie on the circumference of a circle with center O and radius $r$. The angle $\angle AOB = \theta$ radians and $C$ is the midpoint of $OA$. The length $BC = y$ and the arc length $BA = T$. We need to express $y^2$ in terms of $r$ and $\theta$ and then show that when $\theta$ is small, $y^2 = \frac{1}{4}r^2 + \frac{1}{2}r^2\theta^2$. 2. **Express arc length $T$:** The arc $BA$ subtends angle $\theta$ at center $O$, so $$T = r\theta$$ 3. **Coordinates of points:** - Let $O$ be at origin $(0,0)$. - Since $OA = r$, point $A$ lies at $(r,0)$. - Point $C$ is midpoint of $OA$, so $$C = \left(\frac{r}{2}, 0\right)$$ - Point $B$ lies on the circle subtending angle $\theta$ at $O$, so $$B = (r\cos \theta, r\sin \theta)$$ 4. **Length $BC$ calculation:** Distance formula: $$y^2 = (x_B - x_C)^2 + (y_B - y_C)^2 = \left(r\cos \theta - \frac{r}{2}\right)^2 + \left(r\sin \theta - 0\right)^2$$ Simplify: $$y^2 = r^2\left(\cos \theta - \frac{1}{2}\right)^2 + r^2 \sin^2 \theta$$ Expand: $$y^2 = r^2\left(\cos^2 \theta - \cos \theta + \frac{1}{4}\right) + r^2 \sin^2 \theta$$ Group terms: $$y^2 = r^2\left(\cos^2 \theta + \sin^2 \theta - \cos \theta + \frac{1}{4} \right)$$ Use $\cos^2 \theta + \sin^2 \theta = 1$: $$y^2 = r^2\left(1 - \cos \theta + \frac{1}{4}\right) = r^2\left(\frac{5}{4} - \cos \theta \right)$$ 5. **Simplify for small $\theta$:** For small $\theta$, use Taylor expansions: $$\cos \theta \approx 1 - \frac{\theta^2}{2}$$ Substitute: $$y^2 \approx r^2\left(\frac{5}{4} - \left(1 - \frac{\theta^2}{2}\right)\right) = r^2\left(\frac{5}{4} - 1 + \frac{\theta^2}{2}\right) = r^2\left(\frac{1}{4} + \frac{\theta^2}{2}\right)$$ 6. **Final result:** $$\boxed{y^2 = \frac{1}{4}r^2 + \frac{1}{2}r^2 \theta^2}$$ which matches the required expression when $\theta$ is small.