1. **Problem Statement:** We are given a right triangle ABC with a right angle at B. Points C, B, and D lie on a horizontal line with CB = 8 cm and BD = 12 cm. We need to find the length of AD. 2. **Understanding the figure:** Since CD is horizontal and B lies between C and D, the length CD = CB + BD = 8 + 12 = 20 cm. 3. **Identifying segments:** Point A is vertically above B, so AB is perpendicular to CD. Since AB is vertical and CD is horizontal, triangle ABC is right-angled at B. 4. **Using the Pythagorean theorem:** To find AD, note that AD = AB + BD because A is above B and D is on the horizontal line. 5. **Finding AB:** We know AC is the hypotenuse of triangle ABC. Using Pythagoras: $$AC^2 = AB^2 + BC^2$$ But the problem does not give AC directly. However, since the problem asks for AD and gives options, we can infer AB from the right triangle. 6. **Calculate AB:** Since CB = 8 cm and BD = 12 cm, and CD = 20 cm, if we consider triangle ABD, where AB is vertical and BD is horizontal, then AD is the hypotenuse of triangle ABD. 7. **Calculate AD:** Using Pythagoras in triangle ABD: $$AD^2 = AB^2 + BD^2$$ We need AB to calculate AD. Since AB is vertical from B to A, and AB is the height of triangle ABC, and BC = 8 cm, if AC is the hypotenuse, then: Assuming AC = 10 cm (from option a), check if it fits: $$10^2 = AB^2 + 8^2$$ $$100 = AB^2 + 64$$ $$AB^2 = 36$$ $$AB = 6$$ 8. **Now calculate AD:** $$AD^2 = AB^2 + BD^2 = 6^2 + 12^2 = 36 + 144 = 180$$ $$AD = \sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$$ 9. **Answer:** The length of AD is $6\sqrt{5}$ cm, which corresponds to option (c). **Final answer:** $AD = 6\sqrt{5}$ cm.