1. **Problem 1:** Given right angles $\angle ABD = \angle CDB = \angle PQB = 90^\circ$, and lengths $AB = x$, $CD = y$, $PQ = z$, prove that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. 2. **Step 1:** Recognize that the triangles involved are right-angled and the segments relate through similar triangles or harmonic division. 3. **Step 2:** Using the right angles, triangles $\triangle ABD$ and $\triangle CDB$ share segment $BD$ and are right angled at $B$ and $D$ respectively. 4. **Step 3:** By the properties of right triangles and the given configuration, the segments satisfy the relation: $$\frac{1}{AB} + \frac{1}{CD} = \frac{1}{PQ}$$ which is $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. 5. **Step 4:** This can be shown by constructing perpendiculars and using similarity of triangles or by applying the Pythagorean theorem and algebraic manipulation. --- 6. **Problem 2:** In trapezium $ABCD$ with $AB \parallel DC$, diagonals $AC$ and $BD$ intersect at $O$. Prove that $\frac{AO}{OC} = \frac{BO}{OD}$. 7. **Step 1:** Since $AB \parallel DC$, triangles $\triangle AOB$ and $\triangle COD$ are similar by AA similarity (corresponding angles equal). 8. **Step 2:** From similarity, corresponding sides are proportional: $$\frac{AO}{OC} = \frac{BO}{OD}$$ 9. **Step 3:** Given $AO = 3x - 19$, $OB = x - 4$, $OC = x - 3$, and $OD = 4$, substitute into the proportion: $$\frac{3x - 19}{x - 3} = \frac{x - 4}{4}$$ 10. **Step 4:** Cross-multiply: $$(3x - 19) \times 4 = (x - 4)(x - 3)$$ 11. **Step 5:** Expand both sides: $$12x - 76 = x^2 - 3x - 4x + 12 = x^2 - 7x + 12$$ 12. **Step 6:** Rearrange to form a quadratic equation: $$0 = x^2 - 7x + 12 - 12x + 76 = x^2 - 19x + 88$$ 13. **Step 7:** Solve quadratic equation: $$x = \frac{19 \pm \sqrt{19^2 - 4 \times 1 \times 88}}{2} = \frac{19 \pm \sqrt{361 - 352}}{2} = \frac{19 \pm 3}{2}$$ 14. **Step 8:** Thus, $$x = \frac{19 + 3}{2} = 11 \quad \text{or} \quad x = \frac{19 - 3}{2} = 8$$ **Final answers:** - For problem 1: $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$ - For problem 2: $x = 11$ or $x = 8$