1. **Stating the problem:** We have a right triangle with angles A and C, and sides labeled as follows: side opposite angle A is $a$, side opposite angle C is $c$, and the hypotenuse is $b$. We want to understand the relationships between the sides and the tangents of angles A and C. 2. **Formulas and important rules:** The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. For angle A: $$\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} = \frac{a}{c}$$ From this, we can express $a$ in terms of $c$ and $\tan A$: $$a = c \cdot \tan A$$ For angle C: $$\tan C = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} = \frac{c}{a}$$ From this, we can express $c$ in terms of $a$ and $\tan C$: $$c = a \cdot \tan C$$ 3. **Intermediate work and explanation:** - The tangent function relates the sides opposite and adjacent to the angle. - Using the tangent of angle A, we find $a$ by multiplying $c$ by $\tan A$. - Using the tangent of angle C, we find $c$ by multiplying $a$ by $\tan C$. 4. **Summary:** The key relationships are: $$a = c \cdot \tan A$$ $$c = a \cdot \tan C$$ These formulas allow you to find one side length if you know the other side length and the tangent of the corresponding angle. This completes the explanation of the tangent relationships in the right triangle.