Math Problem Statement

find the coordinates of point P along the directed line segment AB from A(-2,-8) to B(3,-3) so that the ratio of AP to PB is 3 to 2

Solution

To find the coordinates of point PP along the directed line segment ABAB from A(2,8)A(-2,-8) to B(3,3)B(3,-3) such that the ratio of APAP to PBPB is 3:23:2, you can use the section formula.

The section formula for a point P(x,y)P(x, y) dividing a line segment joining two points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) in the ratio m:nm:n is given by:

P(x,y)=(mx2+nx1m+n,my2+ny1m+n)P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)

Given:

  • A(x1,y1)=(2,8)A(x_1, y_1) = (-2, -8)
  • B(x2,y2)=(3,3)B(x_2, y_2) = (3, -3)
  • Ratio m:n=3:2m:n = 3:2

Substituting the values into the section formula:

P(x,y)=(33+2(2)3+2,3(3)+2(8)3+2)P(x, y) = \left( \frac{3 \cdot 3 + 2 \cdot (-2)}{3+2}, \frac{3 \cdot (-3) + 2 \cdot (-8)}{3+2} \right)

Calculating the coordinates of PP:

x=945=55=1x = \frac{9 - 4}{5} = \frac{5}{5} = 1

y=9165=255=5y = \frac{-9 - 16}{5} = \frac{-25}{5} = -5

So, the coordinates of point PP are (1,5)\mathbf{(1, -5)}.

Would you like further details on this solution or have any questions?

Here are some related questions:

  1. How would the solution change if the ratio was 2:32:3 instead of 3:23:2?
  2. What are the coordinates of the midpoint of segment ABAB?
  3. How do you find the distance between points AA and BB?
  4. Can the section formula be used in three dimensions?
  5. What is the geometric interpretation of the section formula?

Tip: The section formula is a powerful tool to find points that divide a segment in any given ratio, and it can be extended to three dimensions as well.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Section Formula
Ratio and Proportion

Formulas

Section formula for division of a line segment

Theorems

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Suitable Grade Level

Grades 9-12