1. **State the problem:** We have a right triangle with a base of $\frac{5}{4}$ cm, one leg of $\frac{3}{4}$ cm, and a hypotenuse of 1 cm. We need to find the area of the triangle. 2. **Identify the height $h$:** The height $h$ is perpendicular to the base and forms a right angle. We can use the Pythagorean theorem to find $h$ because the triangle is right-angled. 3. **Use the Pythagorean theorem:** For the right triangle with legs $\frac{3}{4}$ cm and $h$, and hypotenuse 1 cm, the theorem states: $$\left(\frac{3}{4}\right)^2 + h^2 = 1^2$$ 4. **Calculate $h^2$:** $$\frac{9}{16} + h^2 = 1$$ $$h^2 = 1 - \frac{9}{16} = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}$$ 5. **Find $h$:** $$h = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4}$$ 6. **Calculate the area of the triangle:** The area formula is: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ Substitute the values: $$\text{Area} = \frac{1}{2} \times \frac{5}{4} \times \frac{\sqrt{7}}{4} = \frac{5\sqrt{7}}{32}$$ 7. **Final answer:** $$\boxed{\frac{5\sqrt{7}}{32} \text{ cm}^2}$$