1. **Circle Theorem Problem:** State the problem: In circle $O$, chord $AB$ subtends an angle of $40^\circ$ at point $C$ on the circumference. Find the angle subtended by the same chord $AB$ at the center $O$. 2. Formula and rules: The angle subtended by a chord at the center of a circle is twice the angle subtended at any point on the circumference. 3. Work: $$\angle AOB = 2 \times \angle ACB = 2 \times 40^\circ = 80^\circ$$ 4. Explanation: The angle at the center is always twice the angle at the circumference for the same chord. --- 5. **Collinear Points Problem:** State the problem: Given points $A(1,2)$, $B(3,6)$, and $C(k,10)$, find the value of $k$ such that points $A$, $B$, and $C$ are collinear. 6. Formula and rules: Points are collinear if the slope between $AB$ equals the slope between $BC$. 7. Work: $$\text{slope}_{AB} = \frac{6-2}{3-1} = \frac{4}{2} = 2$$ $$\text{slope}_{BC} = \frac{10-6}{k-3} = \frac{4}{k-3}$$ Set equal: $$2 = \frac{4}{k-3}$$ Multiply both sides: $$2 \times (k-3) = 4$$ $$\cancel{2} \times (k-3) = \cancel{4}$$ $$k-3 = 2$$ $$k = 5$$ 8. Explanation: The value $k=5$ makes the slopes equal, so points are collinear. --- 9. **Orthogonal Vectors Problem:** State the problem: Given vectors $\mathbf{u} = (2, -3)$ and $\mathbf{v} = (x, 4)$, find $x$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal. 10. Formula and rules: Two vectors are orthogonal if their dot product is zero. 11. Work: $$\mathbf{u} \cdot \mathbf{v} = 2 \times x + (-3) \times 4 = 0$$ $$2x - 12 = 0$$ $$2x = 12$$ $$x = 6$$ 12. Explanation: When $x=6$, the dot product is zero, so vectors are orthogonal.