Finding Point P2 Using Midpoint Formula

In coordinate geometry, finding the midpoint between two points is a fundamental concept. The midpoint formula allows us to determine the coordinates of a point that divides the line segment connecting two given points into two equal halves. This article will explore how to find the coordinates of a second point, P2, using the midpoint formula and algebraic methods.

Understanding the Problem

The problem presented is to find the coordinates of point P2 (x2, y2), given that the midpoint between point P1 (3, -6) and point P2 (x2, y2) is -2, 3. The midpoint formula is a powerful tool in coordinate geometry, and it states that the coordinates of the midpoint M between two points (x1, y1) and (x2, y2) are given by:

Midpoint Formula: [ M left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right) ]

Substituting the given values into the midpoint formula, we can set up two equations to solve for the coordinates of P2.

Algebraic Solution

Given the midpoint is -2, 3:

Using the x-coordinate of the midpoint:

The x-coordinate of the midpoint is given by:

[ frac{3 x_2}{2} -2 ]

Multiplying both sides by 2:

[ 3 x_2 -4 ]

Solving for x2 by subtracting 3 from both sides:

[ x_2 -7 ]

So, the x-coordinate of P2 is -7.

Using the y-coordinate of the midpoint:

The y-coordinate of the midpoint is given by:

[ frac{-6 y_2}{2} 3 ]

Multiplying both sides by 2:

[ -6 y_2 6 ]

Solving for y2 by adding 6 to both sides:

[ y_2 12 ]

So, the y-coordinate of P2 is 12.

Therefore, the coordinates of point P2 are (-7, 12).

Verification

An easy way to verify the solution is to calculate the midpoint between P1 (3, -6) and P2 (-7, 12). Using the midpoint formula:

[ M left(frac{3 (-7)}{2}, frac{-6 12}{2}right) left(frac{-4}{2}, frac{6}{2}right) (-2, 3) ]

The calculated midpoint is indeed (-2, 3), confirming that our solution is correct.

Additionally, moving from P3 (-6, -3) to P2 (-7, 12), we observe that we move 5 units to the left and 9 units upwards, which aligns with the movement from -2, 3 to -7, 12.

Conclusion

By applying the midpoint formula and algebraic methods, we can accurately determine the coordinates of point P2. The coordinates of P2 are (-7, 12).

Understanding and utilizing the midpoint formula is crucial in coordinate geometry, as it provides a straightforward method to solve problems involving the midpoint of line segments. Whether using the algebraic approach or verifying by calculating the midpoint directly, the solution remains consistent and reliable.