1. **Problem statement:** We have a right triangle ABC with a right angle at B. The side AB is 12 cm, and the hypotenuse AC is 13 cm. We need to find the length of side BC. 2. **Formula used:** In a right triangle, by the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the other two sides: $$AC^2 = AB^2 + BC^2$$ 3. **Apply the values:** Substitute the known lengths: $$13^2 = 12^2 + BC^2$$ 4. **Calculate squares:** $$169 = 144 + BC^2$$ 5. **Isolate $BC^2$:** $$BC^2 = 169 - 144 = 25$$ 6. **Find BC:** $$BC = \sqrt{25} = 5$$ 7. **Answer:** The length of side BC is 5 cm. --- **Next, for the trigonometric functions of the angles adjacent to the right angle:** (a) **sin θ** is the ratio of the length of the side opposite the angle to the hypotenuse. (b) **cos θ** is the ratio of the length of the side adjacent to the angle to the hypotenuse. (c) **tan θ** is the ratio of the length of the side opposite the angle to the side adjacent to the angle. For example, if θ is angle A, then: - Opposite side is BC = 5 - Adjacent side is AB = 12 - Hypotenuse is AC = 13 So: $$\sin A = \frac{5}{13}$$ $$\cos A = \frac{12}{13}$$ $$\tan A = \frac{5}{12}$$ Similarly, for angle C: - Opposite side is AB = 12 - Adjacent side is BC = 5 - Hypotenuse is AC = 13 So: $$\sin C = \frac{12}{13}$$ $$\cos C = \frac{5}{13}$$ $$\tan C = \frac{12}{5}$$