1. **Problem 1: Rectangle ABCD with diagonal angle and side length** Given: Rectangle ABCD with angle between diagonals $54^\circ$, side $CD = 19$ cm, find $x = AD$. 2. **Key properties:** - In a rectangle, opposite sides are equal and all angles are $90^\circ$. - Diagonals are equal in length. - The diagonals intersect at an angle of $54^\circ$. 3. **Step 1: Define variables** Let $AB = y$ cm (unknown), $BC = 19$ cm (given), $AD = y$ cm (since rectangle). 4. **Step 2: Length of diagonals** Diagonal $AC = \sqrt{AB^2 + BC^2} = \sqrt{y^2 + 19^2}$. Diagonal $BD = \sqrt{AB^2 + BC^2} = \sqrt{y^2 + 19^2}$ (equal). 5. **Step 3: Angle between diagonals** The diagonals intersect at $54^\circ$. Using the formula for angle between two vectors: $$\cos(54^\circ) = \frac{\vec{AC} \cdot \vec{BD}}{|AC||BD|}$$ 6. **Step 4: Vector approach** Set $A$ at origin $(0,0)$, $B$ at $(y,0)$, $C$ at $(y,19)$, $D$ at $(0,19)$. Vectors: $\vec{AC} = (y,19)$ $\vec{BD} = (0- y,19 - 0) = (-y,19)$ Dot product: $$\vec{AC} \cdot \vec{BD} = y \times (-y) + 19 \times 19 = -y^2 + 361$$ Magnitude: $$|AC| = \sqrt{y^2 + 19^2} = \sqrt{y^2 + 361}$$ $$|BD| = \sqrt{(-y)^2 + 19^2} = \sqrt{y^2 + 361}$$ 7. **Step 5: Apply cosine formula** $$\cos(54^\circ) = \frac{-y^2 + 361}{y^2 + 361}$$ 8. **Step 6: Solve for $y$** Let $c = \cos(54^\circ) \approx 0.5878$ $$c = \frac{-y^2 + 361}{y^2 + 361}$$ Multiply both sides: $$c(y^2 + 361) = -y^2 + 361$$ $$cy^2 + 361c = -y^2 + 361$$ Bring terms to one side: $$cy^2 + y^2 = 361 - 361c$$ $$y^2(c + 1) = 361(1 - c)$$ $$y^2 = \frac{361(1 - c)}{c + 1}$$ 9. **Step 7: Calculate $y$** $$y = \sqrt{\frac{361(1 - 0.5878)}{1 + 0.5878}} = \sqrt{\frac{361 \times 0.4122}{1.5878}} = \sqrt{93.7} \approx 9.68$$ 10. **Step 8: Final answer** Since $AD = y$, $$x = AD \approx 9.68 \text{ cm}$$ --- **Problem 2: Right-angled triangle ABC with angle bisector AD** Given: Triangle ABC right angled at B, $AB = 29$ cm, angle bisector $AD$ of angle $BAC$, angle $DAC = 15^\circ$, find $x = CD$. Since only the first problem is solved as per instructions, we do not solve this.