Problem 4 - The Calculus with Analytic Geometry

Question

Derive midpoint formulas (2) and (3) if \(P_{1}\left(x_{1}, y_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}\right)\) are both in the second quadrant and \(x_{2}>x_{1}\) and \(y_{1}>y_{2}\).

Step-by-Step Solution

Answer

The midpoint is \( \big( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \big) \).

Step 1 - Identify the Coordinates

Identify the coordinates of the points given: Point 1: \(P_{1}\big(x_{1}, y_{1}\big)\) Point 2: \(P_{2}\big(x_{2}, y_{2}\big)\)

Step 2 - Verify Quadrant

Ensure both points lie in the second quadrant where \(x < 0\) and \(y > 0\). Given conditions are \(x_{2}>x_{1}\) and \(y_{1}>y_{2}\), both negative values for \(x\) and positive values for \(y\) should still satisfy this.

Step 3 - Apply the Midpoint Formula

Use the midpoint formula to derive the midpoint coordinates between the two points. The midpoint \(M(x, y)\) of \(P_{1}\big(x_{1}, y_{1}\big)\) and \(P_{2}\big(x_{2}, y_{2}\big)\) is given by:\[M\big(x, y\big) = \bigg( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}} {2} \bigg)\]

Step 4 - Substitute and Simplify

Substitute the coordinates of \(P_{1}\) and \(P_{2}\) into the midpoint formula:\[ x = \frac{x_{1} + x_{2}}{2} \text{ and } y = \frac{y_{1} + y_{2}}{2}\]Since \(x_{2} > x_{1}\) and \(y_{1} > y_{2}\), it follows naturally that the x-coordinate of the midpoint will be slightly less than \(x_{2}\) and the y-coordinate will be slightly more than \(y_{2}\).

Step 5 - Confirm Formulas

The final expressions for the midpoint coordinates are:\[ \boxed{ \(x_M = \frac{x_{1} + x_{2}}{2} \text{ and } y_M = \frac{y_{1} + y_{2}}{2} \) }\]This represents the midpoint formula applicable to any two points in the plane including those in the second quadrant.

Key Concepts

Quadrants in Coordinate Geometry

To understand the midpoint formula, it's crucial to grasp the concept of quadrants in coordinate geometry. The coordinate plane is divided into four quadrants defined by the x-axis (horizontal) and y-axis (vertical). Quadrants are numbered counterclockwise starting from the upper right.

• First Quadrant (I): Here, both x and y coordinates are positive.
• Second Quadrant (II): x is negative and y is positive.
• Third Quadrant (III): Both x and y are negative.
• Fourth Quadrant (IV): x is positive and y is negative.

Knowing the specific quadrant of points helps us predict the signs of the coordinates. In this problem, both points lie in the second quadrant, meaning their x-values are negative, and y-values are positive. This lays out the foundation for applying the midpoint formula correctly.

Midpoint Coordinates

Midpoint coordinates are critical in geometry, helping determine the exact middle point between two points. If you have two points, say \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\), you calculate the midpoint to find a point exactly between them on a line.
To find the midpoint \(M(x, y)\), use the formula:
\[M(x, y) = \bigg( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \bigg)\]
This means you take the average of the x-coordinates and the average of the y-coordinates. This formula does not change regardless of which quadrant the points are in. It provides a systematic way to determine the midpoint's coordinates.

Step-by-Step Mathematical Proof

Deriving the midpoint formula requires a step-by-step approach. Here's a detailed breakdown, starting with the fundamental elements:
Step 1: Identify Coordinates
Acknowledge your points as \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\).
Step 2: Verify Quadrant
Ensure these points are in the correct quadrant; here, they should be in the second quadrant with \(x_1, x_2 < 0\) and \(y_1, y_2 > 0\).
Step 3: Apply Midpoint Formula
The midpoint is found using \[M(x, y) = \bigg(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}\bigg)\]
Step 4: Substitute Values
Replace \(x_1, x_2, y_1, y_2\) in the formula:
\[M(x, y) = \bigg(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y{2}}{2}\bigg)\]
Since \(x_2 > x_1\) and \(y_1 > y_2\), the calculations follow naturally.
Step 5: Confirm Formulas
The final midpoint coordinates are \[x_M = \frac{x_{1} + x_{2}}{2}, y_M = \frac{y_{1} + y_{2}}{2}\]
Thus, you get the midpoint irrespective of points location in the second quadrant.