1. **Problem:** Find the midpoint of the line segment with endpoints A(-4,5) and B(-5,0). 2. **Formula:** Midpoint $M$ of segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is given by: $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ 3. **Calculation:** $$M = \left(\frac{-4 + (-5)}{2}, \frac{5 + 0}{2}\right) = \left(\frac{-9}{2}, \frac{5}{2}\right) = (-4.5, 2.5)$$ 4. **Answer:** The midpoint is $\boxed{D\ (-4.5, 2.5)}$. 5. **Problem:** Find the length of the line segment with endpoints C(-3, -5) and D(2, 4). 6. **Formula:** Distance $d$ between points $(x_1,y_1)$ and $(x_2,y_2)$ is: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 7. **Calculation:** $$d = \sqrt{(2 - (-3))^2 + (4 - (-5))^2} = \sqrt{(5)^2 + (9)^2} = \sqrt{25 + 81} = \sqrt{106}$$ 8. **Answer:** The length is $\boxed{\sqrt{106}}$ (option C). 9. **Problem:** Find an equation for the circle with centre $(0,0)$ and radius $1$. 10. **Formula:** Equation of circle with centre $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 11. **Calculation:** Here $h=0$, $k=0$, $r=1$, so: $$ x^2 + y^2 = 1^2 = 1 $$ 12. **Answer:** The equation is $\boxed{x^2 + y^2 = 1}$ (option C). 13. **Problem:** The intersection of all medians of a triangle is called? 14. **Answer:** The intersection point of medians is called the $\boxed{\text{Centroid}}$ (option C). 15. **Problem:** The point $(-1,2)$ lies on a circle with centre $(0,0)$. Find the equation of the circle. 16. **Formula:** Use $x^2 + y^2 = r^2$ and find $r$ by distance from centre to point: $$r = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$$ 17. **Equation:** $$x^2 + y^2 = 5$$ 18. **Answer:** The equation is $\boxed{x^2 + y^2 = 5}$ (option B). 19. **Problem:** The quadrilateral formed by the midpoints of any quadrilateral is? 20. **Answer:** It is always a $\boxed{\text{Parallelogram}}$ (option D). 21. **Problem:** Determine the shortest distance between point $D(2,5)$ and line $2y - 3x = 5$. 22. **Formula:** Distance $d$ from point $(x_0,y_0)$ to line $Ax + By + C = 0$ is: $$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$ 23. **Rewrite line:** $$2y - 3x = 5 \implies -3x + 2y - 5 = 0$$ 24. **Calculate numerator:** $$|-3(2) + 2(5) - 5| = |-6 + 10 - 5| = |-1| = 1$$ 25. **Calculate denominator:** $$\sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$$ 26. **Distance:** $$d = \frac{1}{\sqrt{13}}$$ 27. **Answer:** The shortest distance is $\boxed{\frac{1}{\sqrt{13}}}$. **Final answers:** 1: D $(-4.5, 2.5)$ 2: C $\sqrt{106}$ 3: C $x^2 + y^2 = 1$ 4: C Centroid 5: B $x^2 + y^2 = 5$ 6: D Parallelogram 7: $\frac{1}{\sqrt{13}}$