Partitioning Line Segments via Ratio - Geometry

1. Find $P$ on $A(2,-4)$ to $B(8,2)$ such that $AP:PB=1:2$ internally.

   $P(4,-2)$. Apply section formula with ratio $1:2$ to both coordinates.

2. Find $P$ on $A(0,5)$ to $B(8,1)$ such that $AP:PB=3:1$ internally.

   $P(6,2)$. Apply section formula with ratio $3:1$ to both coordinates.

3. Identify the ratio $m:n$ that produces the midpoint of a segment using the section formula.

   $1:1$. Equal division creates the midpoint of any segment.

4. Find the point dividing $A(4,0)$ to $B(0,8)$ in ratio $3:1$ internally.

   $P(1,6)$. Apply section formula with ratio $3:1$ to both coordinates.

5. Find the point dividing directed segment $A(1,1)$ to $B(5,3)$ in ratio $2:1$ externally.

   $P(9,5)$. Apply external section formula with ratio $2:1$ to both coordinates.

6. State the section formula for point $P$ dividing directed segment $A(x_1,y_1)$ to $B(x_2,y_2)$ in ratio $m:n$ (internal).

   $P\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)$. Weighted average formula for internal division where weights are $n$ and $m$.

7. Find the point dividing $A(2,2)$ to $B(8,8)$ in ratio $1:2$ internally.

   $P(4,4)$. Apply section formula with both coordinates using ratio $1:2$.

8. Which condition on $t$ makes $P$ on the ray opposite $B$ from $A$ along line $AB$?

   $t<0$. Negative parameter places point on opposite side of $A$.

9. Find the point dividing $A(0,0)$ to $B(0,12)$ in ratio $1:2$ internally.

   $P(0,4)$. Apply section formula: $\frac{1 \cdot 12 + 2 \cdot 0}{1+2} = 4$.

10. Find $P$ on $A(-4,-1)$ to $B(2,5)$ such that $AP:PB=1:5$ internally.

    $P(-3,0)$. Apply section formula with ratio $1:5$ to both coordinates.

11. Find the point dividing $A(0,0)$ to $B(10,0)$ in ratio $3:2$ internally.

    $P(6,0)$. Apply section formula: $\frac{3 \cdot 10 + 2 \cdot 0}{3+2} = 6$.

12. State the section formula for point $P$ dividing directed segment $A(x_1,y_1)$ to $B(x_2,y_2)$ in ratio $m:n$ (external).

    $P\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\right)$. External division uses subtraction in numerator and denominator.

13. Identify the parameter $t$ if $P$ divides $A$ to $B$ internally in ratio $3:2$ using $P=A+t(B-A)$.

    $t=\frac{3}{5}$. Convert ratio to parameter: $t = \frac{3}{3+2} = \frac{3}{5}$.

14. Find the point $P$ on segment $A(-1,2)$ to $B(5,8)$ with $t=\frac{1}{2}$ in $P=A+t(B-A)$.

    $P(2,5)$. Apply vector formula: $P = (-1,2) + \frac{1}{2}(6,6) = (2,5)$.

15. Find the point dividing $A(5,-3)$ to $B(1,9)$ in ratio $2:1$ internally.

    $P\left(\frac{7}{3},1\right)$. Apply section formula with ratio $2:1$ to both coordinates.

16. Identify the ratio $m:n$ for point $P(4,0)$ on segment from $A(0,0)$ to $B(10,0)$ (internal).

    $2:3$. Distance ratio: $AP:PB = 4:6 = 2:3$.

17. Find $P$ if directed segment $A(0,0)$ to $B(3,3)$ is partitioned with $t=-1$ in $P=A+t(B-A)$.

    $P(-3,-3)$. Negative parameter extends point beyond $A$ on the line.

18. Identify the ratio $m:n$ for point $P(2,2)$ on segment from $A(0,0)$ to $B(6,6)$ (internal).

    $1:2$. Distance ratio: $AP:PB = 2:4 = 1:2$.

19. Which point is closer to $B$ when $P$ divides $\overline{AB}$ internally in ratio $m:n$ with $m>n$?

    $P$ is closer to $B$. Larger first ratio means point is closer to the second endpoint.

20. Identify the distance relationship for internal division ratio $m:n$ on $\overline{AB}$ in terms of $AP$ and $PB$.

    $\frac{AP}{PB}=\frac{m}{n}$. Internal division preserves the ratio of distances.

21. Find $t$ if $P$ is the midpoint of directed segment $AB$ in the form $P=A+t(B-A)$.

    $t=\frac{1}{2}$. Midpoint corresponds to $t = \frac{1}{2}$ in the vector formula.

22. Find the point dividing $A(-6,2)$ to $B(3,11)$ in ratio $2:1$ internally.

    $P(0,8)$. Apply section formula with ratio $2:1$ to both coordinates.

23. Which point is closer to $A$ when $P$ divides $\overline{AB}$ internally in ratio $m:n$ with $m<n$?

    $P$ is closer to $A$. Smaller first ratio means point is closer to the first endpoint.

24. Find $P$ if $A(0,0)$, $B(12,6)$, and $AP:PB=1:3$ internally.

    $P(3,\frac{3}{2})$. Apply section formula with ratio $1:3$ to both coordinates.

25. State the coordinate form of $P=A+t(B-A)$ when $A(x_1,y_1)$ and $B(x_2,y_2)$ are given.

    $P(x_1+t(x_2-x_1),\ y_1+t(y_2-y_1))$. Coordinate form of the vector parametric equation.

26. Find $P$ if $A(6,2)$, $B(-2,10)$, and $AP:PB=3:1$ internally.

    $P(0,8)$. Apply section formula with ratio $3:1$ to both coordinates.

27. Find the point $P$ on segment $A(10,0)$ to $B(4,12)$ with $t=\frac{2}{3}$ in $P=A+t(B-A)$.

    $P(6,8)$. Apply vector formula: $P = (10,0) + \frac{2}{3}(-6,12) = (6,8)$.

28. Find the point $P$ on segment $A(3,1)$ to $B(9,7)$ with $t=\frac{1}{3}$ in $P=A+t(B-A)$.

    $P(5,3)$. Apply vector formula: $P = (3,1) + \frac{1}{3}(6,6) = (5,3)$.

29. Find the point dividing $A(0,0)$ to $B(10,0)$ in ratio $1:4$ internally.

    $P(2,0)$. Apply section formula: $\frac{1 \cdot 10 + 4 \cdot 0}{1+4} = 2$.

30. Find $P$ if $A(-3,9)$, $B(9,-3)$, and $AP:PB=1:2$ internally.

    $P(1,5)$. Apply section formula with ratio $1:2$ to both coordinates.

31. Identify the ratio $m:n$ for point $P(6,0)$ on segment from $A(2,0)$ to $B(10,0)$ (internal).

    $1:1$. Distance ratio: $AP:PB = 4:4 = 1:1$.

32. Identify the denominator that must be nonzero in the external section formula for ratio $m:n$.

    $m-n\ne 0$. External division is undefined when $m = n$.

33. Find the point dividing $A(0,0)$ to $B(6,9)$ in ratio $2:1$ internally.

    $P(4,6)$. Apply section formula with ratio $2:1$ to both coordinates.

34. Find the point dividing $A(2,5)$ to $B(8,11)$ in ratio $1:2$ internally.

    $P(4,7)$. Apply section formula with ratio $1:2$ to both coordinates.

35. Find $P$ if directed segment $A(1,2)$ to $B(4,8)$ is partitioned with $t=2$ in $P=A+t(B-A)$.

    $P(7,14)$. Parameter greater than 1 extends point beyond $B$.

36. Find the point dividing $A(1,7)$ to $B(9,-1)$ in ratio $1:1$ internally.

    $P(5,3)$. Ratio $1:1$ gives the midpoint formula result.

37. Find $P$ on $A(7,7)$ to $B(1,-5)$ such that $AP:PB=2:1$ internally.

    $P(3,-1)$. Apply section formula with ratio $2:1$ to both coordinates.

38. Which condition on $t$ in $P=A+t(B-A)$ guarantees that $P$ lies between $A$ and $B$?

    $0<t<1$. Point lies between endpoints when parameter is between 0 and 1.

39. Identify the key idea of the section formula: $P$ is a weighted average of which points?

    $A(x_1,y_1)$ and $B(x_2,y_2)$. Section formula creates weighted averages of the endpoint coordinates.

40. Find the point dividing $A(-4,0)$ to $B(6,0)$ in ratio $2:3$ internally.

    $P(0,0)$. Apply section formula: $\frac{2 \cdot 6 + 3 \cdot (-4)}{2+3} = 0$.

41. Find the point dividing $A(-2,-2)$ to $B(4,10)$ in ratio $1:2$ internally.

    $P(0,2)$. Apply section formula with ratio $1:2$ to both coordinates.

42. Which condition on $t$ makes $P$ beyond $B$ on the ray from $A$ through $B$?

    $t>1$. Point extends beyond $B$ when parameter exceeds 1.

43. State the vector form for the point $P$ that is fraction $t$ of the way from $A$ to $B$ on a directed segment.

    $P=A+t(B-A)$. Vector form where $t$ is the fractional distance from $A$ to $B$.

44. What is the relationship between parameter $t$ and ratio $m:n$ for internal division from $A$ toward $B$?

    $m:n=t:(1-t)$. Ratio converts to parameter by using $t$ and its complement.

45. Find the point dividing directed segment $A(0,0)$ to $B(4,0)$ in ratio $2:1$ externally.

    $P(8,0)$. Apply external section formula: $\frac{2 \cdot 4 - 1 \cdot 0}{2-1} = 8$.

46. Identify the parameter $t$ if $P$ divides $A$ to $B$ internally in ratio $1:4$ using $P=A+t(B-A)$.

    $t=\frac{1}{5}$. Convert ratio to parameter: $t = \frac{1}{1+4} = \frac{1}{5}$.

47. Find the point dividing $A(-2,4)$ to $B(6,0)$ in ratio $3:1$ internally.

    $P(4,1)$. Apply section formula with ratio $3:1$ to both coordinates.

48. Find the point dividing directed segment $A(0,0)$ to $B(6,0)$ in ratio $1:2$ externally.

    $P(-6,0)$. Apply external section formula: $\frac{1 \cdot 6 - 2 \cdot 0}{1-2} = -6$.

49. Identify the ratio $m:n$ for point $P(3,0)$ on segment from $A(-1,0)$ to $B(7,0)$ (internal).

    $1:1$. Distance ratio: $AP:PB = 4:4 = 1:1$.

50. What is the relationship between ratio $m:n$ (internal) and parameter $t$ in $P=A+t(B-A)$?

    $t=\frac{m}{m+n}$. Parameter $t$ equals the first part over the total.

51. What is the midpoint formula as a special case of partitioning $A(x_1,y_1)$ to $B(x_2,y_2)$?

    $M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Special case when $m=n=1$ in the section formula.

52. Find the point dividing $A(0,6)$ to $B(12,0)$ in ratio $1:3$ internally.

    $P(3,\frac{9}{2})$. Apply section formula with ratio $1:3$ to both coordinates.

53. Find the point dividing directed segment $A(2,0)$ to $B(8,6)$ in ratio $1:2$ externally.

    $P(-4,-6)$. Apply external section formula with ratio $1:2$ to both coordinates.

54. Find $P$ if $A(2,4)$, $B(14,10)$, and $AP:PB=2:4$ internally (simplify ratio first).

    $P(6,6)$. Simplify $2:4$ to $1:2$ then apply section formula.

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