1. **State the problem:** Given a circle with radius $7$ cm and diameter $TC$, find the circumference, area, various angle measures, arc measures, arc lengths, and the area of the shaded region. 2. **Formulas and rules:** - Circumference of a circle: $$C = 2\pi r$$ - Area of a circle: $$A = \pi r^2$$ - Arc length: $$s = \frac{\theta}{360} \times 2\pi r$$ where $\theta$ is the arc measure in degrees. - Area of a sector: $$A_{sector} = \frac{\theta}{360} \times \pi r^2$$ - Central angle equals the measure of the intercepted arc. - Inscribed angle measure is half the measure of its intercepted arc. 3. **Calculate circumference:** $$C = 2\pi \times 7 = 14\pi$$ 4. **Calculate area:** $$A = \pi \times 7^2 = 49\pi$$ 5. **Find $m\angle ART$:** Given as $26^\circ$. 6. **Find $m$ of arc $PC$:** Given $\angle ARP = 77^\circ$ is an inscribed angle intercepting arc $PC$, so $$m\widehat{PC} = 2 \times 77 = 154^\circ$$ 7. **Find $m$ of arc $AP$:** Since $TC$ is diameter, semicircle $TPC$ is $180^\circ$. Arc $AP$ is the remainder of semicircle minus arc $PC$: $$m\widehat{AP} = 180 - 154 = 26^\circ$$ 8. **Find $m$ of arc $PAT$:** Given arc $AT = 32^\circ$, and arc $AP = 26^\circ$, so $$m\widehat{PAT} = m\widehat{AP} + m\widehat{AT} = 26 + 32 = 58^\circ$$ 9. **Find $m\angle ACP$:** $\angle ACP$ is inscribed angle intercepting arc $AP$, so $$m\angle ACP = \frac{1}{2} \times m\widehat{AP} = \frac{1}{2} \times 26 = 13^\circ$$ 10. **Find $m\angle ACT$:** $\angle ACT$ intercepts arc $AT$, so $$m\angle ACT = \frac{1}{2} \times 32 = 16^\circ$$ 11. **Find $m$ of arc $CE$:** Given arc $TL = 65^\circ$ and $TC$ is diameter, arc $CE$ is remainder of circle minus arcs $TL$ and $AT$: $$m\widehat{CE} = 360 - (65 + 32) = 263^\circ$$ 12. **Find $m\angle LET$:** Inscribed angle intercepting arc $CE$, so $$m\angle LET = \frac{1}{2} \times 263 = 131.5^\circ$$ 13. **Find $m$ of arc $LE$:** Since $L$ and $E$ lie on arc $CE$, and $m\widehat{CE} = 263^\circ$, arc $LE$ is part of $CE$ but no exact measure given, so cannot determine without more info. 14. **Find $m\angle LPE$:** No data to determine this angle. 15. **Find $m$ of arc $TLC$:** Sum of arcs $TL$ and $LC$. Arc $TL = 65^\circ$, arc $LC$ is part of arc $CE$ but unknown, so cannot determine. 16. **Arc length of $AT$:** $$s_{AT} = \frac{32}{360} \times 2\pi \times 7 = \frac{32}{360} \times 14\pi = \frac{32 \times 14\pi}{360} = \frac{448\pi}{360} = \frac{28\pi}{22.5} \approx 3.91$$ cm 17. **Arc length of $TL$:** $$s_{TL} = \frac{65}{360} \times 14\pi = \frac{910\pi}{360} = \frac{91\pi}{36} \approx 7.94$$ cm 18. **Area of shaded region (sector $AT$):** $$A_{shaded} = \frac{32}{360} \times 49\pi = \frac{1568\pi}{360} = \frac{98\pi}{22.5} \approx 13.69$$ cm$^2$ **Final answers:** - Circumference = $14\pi$ cm - Area = $49\pi$ cm$^2$ - $m\angle ART = 26^\circ$ - $m\widehat{PC} = 154^\circ$ - $m\widehat{AP} = 26^\circ$ - $m\widehat{PAT} = 58^\circ$ - $m\angle ACP = 13^\circ$ - $m\angle ACT = 16^\circ$ - $m\widehat{CE} = 263^\circ$ - $m\angle LET = 131.5^\circ$ - Arc length $AT \approx 3.91$ cm - Arc length $TL \approx 7.94$ cm - Area shaded $\approx 13.69$ cm$^2$ Some requested values cannot be determined with given data.