Introduction

Determining the coordinates of certain vertices in a parallelogram can be readily accomplished using fundamental principles of coordinate geometry. This article explores how to find the value of k and m for a parallelogram defined by given vertices P, Q, R, and S. By leveraging the midpoint theorem, we can find the intersection point of the diagonals and determine the required coordinates.

Understanding the Problem

The given problem involves a parallelogram with vertices P (1, 2), Q (2, -2), R (-1, 0), and S (4, m). We are tasked with finding the values of k and m. Notably, the diagonals of a parallelogram bisect each other, which allows us to use the midpoint formula to solve for these unknowns.

Solving for Point O (Midpoint of Diagonals)

Point O is the intersection point of the diagonals PR and QS, and it is also the midpoint of both diagonals. Using the midpoint formula, where the midpoint (x, y) of a line segment with endpoints (x_1, y_1) and (x_2, y_2) is given by: (x, y) #40;(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}#41;

Applying the Midpoint Formula to Diagonal PR

For diagonal PR with P (1, 2) and R (-1, 0), the midpoint O is calculated as follows:

Calculate the x-coordinate of O:

O_x frac{1 (-1)}{2} frac{0}{2} 0

Calculate the y-coordinate of O:

O_y frac{2 0}{2} frac{2}{2} 1

Therefore, the coordinates of point O for diagonal PR are (0, 1).

Applying the Midpoint Formula to Diagonal QS

For diagonal QS with Q (2, -2) and S (4, m), the midpoint O is calculated as follows:

Calculate the x-coordinate of O:

O_x frac{2 4}{2} frac{6}{2} 3

Calculate the y-coordinate of O:

O_y frac{-2 m}{2}

Since point O is the same for both diagonals, its coordinates must be (0, 1). Therefore, we can set up the following equations:

O_x 0 O_y 1

Solving for h and m

From the midpoint coordinates of diagonal QS, we have:

1 frac{-2 m}{2}

Solving for m by multiplying both sides by 2:

2 -2 m

Adding 2 to both sides:

m 4

For the x-coordinate of point O, we have:

0 frac{2 4}{2}

However, this is a contradiction as the x-coordinate should be 3. Thus, we need to re-evaluate the values of k and m from the proper application of the midpoint formula.

Revisiting the Midpoint Formula Application

Given the correct midpoint (0, 1), we can re-evaluate the coordinates of point S and find the value of k for point P consistently.

For the y-coordinate of S:

1 frac{-2 m}{2}

Substituting m 4:

1 frac{-2 4}{2} frac{2}{2} 1

For the x-coordinate of O:

0 frac{2 4k}{2}

Since O_x is 0, we solve for k:

0 frac{2 4k}{2}

Multiplying both sides by 2:

0 2 4k

Solving for k by subtracting 2 from both sides:

4k -2

Dividing both sides by 4:

k -0.5

Conclusion

The values of k and m for the given parallelogram are k -0.5 and m 4, respectively. These values ensure that the diagonals of the parallelogram bisect each other at the point (0, 1), fulfilling the conditions of a parallelogram.