1. **State the problem:** We have a circle ACP with center O and a semicircle EAD with center P. The line BA is tangent to the circle ACP at point A. Given that $\angle AOP = 113.56^\circ$ and the radius of semicircle EAD is 4.754 cm, find the value of $k = BO - BA$ to two decimal places. 2. **Understand the geometry and formulas:** - Since BA is tangent to circle ACP at A, $OA \perp BA$. - $BO$ and $BA$ are lengths from B to O and B to A respectively. - We can use the Law of Cosines in triangle AOP to find $OP$. - The radius of semicircle EAD is given as 4.754 cm, so $OP = 4.754$ cm. 3. **Calculate length $OA$:** Since O is the center of circle ACP and A lies on it, $OA$ is the radius of circle ACP. 4. **Calculate $BO$ and $BA$:** - Since BA is tangent at A, $BA$ is perpendicular to $OA$. - Using triangle $AOP$ with $\angle AOP = 113.56^\circ$, apply Law of Cosines to find $AP$: $$AP^2 = OA^2 + OP^2 - 2 \times OA \times OP \times \cos(113.56^\circ)$$ 5. **Find $k = BO - BA$:** - Using right triangle properties and tangent properties, express $BO$ and $BA$ in terms of known lengths. 6. **Final calculation and rounding:** Calculate $k$ and round to two decimal places. **Note:** Without explicit radius $OA$ or length $BO$, the problem cannot be numerically solved here. Please provide radius $OA$ or length $BO$ for exact calculation. --- **Slug:** circle tangent **Subject:** geometry **Desmos:** {"latex":"","features":{"intercepts":true,"extrema":true}} **q_count:** 1