1. **Problem statement:** We have a right triangle $\triangle ABD$ with right angle at $B$. The hypotenuse $AD$ is 20 cm. Point $C$ lies on segment $BD$ such that $BC=5$ cm and $CD=11$ cm. We need to find the length of segment $\overline{AC}$. 2. **Known values:** - $AD=20$ cm (hypotenuse) - $BC=5$ cm - $CD=11$ cm - $BD=BC+CD=5+11=16$ cm 3. **Goal:** Find $AC$. 4. **Approach:** Use coordinate geometry or the Pythagorean theorem. Since $\triangle ABD$ is right angled at $B$, we can place points on a coordinate plane for clarity: - Let $B$ be at origin $(0,0)$. - Since $AB$ is vertical, let $A=(0,a)$ for some $a$. - Since $BD$ is horizontal, let $D=(16,0)$ because $BD=16$. 5. **Find $a$ using $AD=20$: $$AD=\sqrt{(16-0)^2+(0 - a)^2}=20$$ $$\sqrt{16^2 + a^2} = 20$$ $$16^2 + a^2 = 20^2$$ $$256 + a^2 = 400$$ $$a^2 = 400 - 256 = 144$$ $$a = 12$$ So, $A=(0,12)$. 6. **Find coordinates of $C$: Since $C$ lies on $BD$ with $BC=5$, and $B=(0,0)$, $C=(5,0)$. 7. **Calculate $AC$: $$AC = \sqrt{(5-0)^2 + (0-12)^2} = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$ **Final answer:** $\boxed{13}$ cm, which corresponds to option B.