1. **Problem statement:** We have a circle with center $C$ and radius 8 cm. The angle $\angle FEG = 45^\circ$ is given. We need to find: a. The measure of $\angle FCG$. b. The length of the chord $FG$ to the nearest tenth of a centimetre. 2. **Understanding the problem and formulas:** - $C$ is the center of the circle, so $CF$ and $CG$ are radii, each 8 cm. - $\angle FEG$ is an inscribed angle subtending chord $FG$. - The central angle $\angle FCG$ subtending the same chord $FG$ is related to the inscribed angle by the rule: the central angle is twice the inscribed angle. Formula for chord length: $$ FG = 2r \sin\left(\frac{\theta}{2}\right) $$ where $r$ is the radius and $\theta$ is the central angle in degrees. 3. **Step a: Find $\angle FCG$** - Since $\angle FEG = 45^\circ$ is an inscribed angle subtending chord $FG$, - The central angle $\angle FCG$ subtending the same chord is: $$ \angle FCG = 2 \times \angle FEG = 2 \times 45^\circ = 90^\circ $$ 4. **Step b: Find length of chord $FG$** - Radius $r = 8$ cm - Central angle $\theta = 90^\circ$ - Using the chord length formula: $$ FG = 2 \times 8 \times \sin\left(\frac{90^\circ}{2}\right) = 16 \times \sin(45^\circ) $$ - We know $\sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071$ - So, $$ FG = 16 \times 0.7071 = 11.3136 \text{ cm} $$ - Rounded to the nearest tenth: $$ FG \approx 11.3 \text{ cm} $$ **Final answers:** - a. $\angle FCG = 90^\circ$ - b. Length of chord $FG \approx 11.3$ cm