1. **Problem statement:** We need to find the value of $x$ in the given right triangle configuration inside the rectangle. 2. **Understanding the problem:** The rectangle has length 25 cm and height 10 cm. The diagonal divides it into two triangles $T_1$ and $T_2$. There is a smaller right triangle with base $x$, height 6 cm, and an angle $\theta$ adjacent to the hypotenuse. 3. **Key formulas:** In a right triangle, the tangent of an angle $\theta$ is the ratio of the opposite side to the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 4. **Applying the tangent formula:** For the smaller right triangle with base $x$ and height 6 cm, $$\tan(\theta) = \frac{6}{x}$$ 5. **Using the larger triangle:** The larger triangle has base $b$ and height 10 cm. Since the two triangles share the angle $\theta$, their tangents are equal: $$\tan(\theta) = \frac{10}{b}$$ 6. **Equate the two expressions for $\tan(\theta)$:** $$\frac{6}{x} = \frac{10}{b}$$ 7. **Solve for $x$:** $$x = \frac{6b}{10} = 0.6b$$ 8. **Find $b$:** The total length of the rectangle is 25 cm, and $b + x = 25$ cm. Substitute $x = 0.6b$: $$b + 0.6b = 25$$ $$1.6b = 25$$ $$b = \frac{25}{1.6} = 15.625$$ 9. **Calculate $x$:** $$x = 0.6 \times 15.625 = 9.375$$ 10. **Final answer:** Rounded to the nearest one decimal place, $$x = 9.4$$ cm.