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The midpoint formula is an essential tool in geometry used to find the center point between two endpoints on a coordinate plane. The midpoint, represented by \( M(x, y) \), is derived from the average of the x-coordinates and y-coordinates of the endpoints. The formula is given by:

\( M_x = \frac{(P_x + Q_x)}{2} \) and \( M_y = \frac{(P_y + Q_y)}{2} \)

This means that to find the midpoint, you add the x-coordinates of the endpoints and divide by 2, and do the same for the y-coordinates. Breaking it down:

• Add the x-coordinates \( P_x \) and \( Q_x \).
• Divide the sum by 2 to get \( M_x \).
• Add the y-coordinates \( P_y \) and \( Q_y \).
• Divide the sum by 2 to get \( M_y \).

In this exercise, the given coordinates of endpoint P are P(1.5, 1.25) and midpoint M is (3, 1). Plugging these into the midpoint formula helps us find the other endpoint Q.

Coordinate geometry, also known as analytic geometry, is a branch of geometry where the positions of points on a plane are described using ordered pairs (x, y). This involves plotting points, lines, and figures on a coordinate plane.

Key concepts include:

• Points: Defined by pairs of coordinates (x, y), which determine their location on the plane.
• Lines: Can be represented by equations, such as y = mx + b (slope-intercept form).
• Distances: Calculated using the distance formula between two points: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
• Midpoints: As explained, calculated using the midpoint formula.

Understanding the coordinates of points is crucial, as demonstrated in the exercise. Endpoint P(1.5, 1.25) and midpoint M(3, 1) were plotted to find the coordinates of the other endpoint Q. This involves solving for unknowns in equations derived from the coordinate positions.

Solving equations is a fundamental skill in mathematics, especially in problems involving coordinate geometry. To find the coordinates of the missing endpoint Q, we need to set up and solve equations using the midpoint formula. Here’s how it’s done:

1. **Set Up Equations:** Use the midpoint coordinates and known endpoint to form equations. For the x-coordinate:

\( 3 = \frac{(1.5 + x)}{2} \)

2. **Solve for x:** Multiply both sides by 2 to isolate x:

\( 3 \times 2 = 1.5 + x \)

Simplify and solve: \( 6 = 1.5 + x \) leads to \( x = 4.5 \. \)

3. **Repeat for y-coordinate:** Follow the same process to find y:

\ 1 = \frac{(1.25 + y)} \{2} \

Multiply both sides by 2: \ 1 \times 2 = 1.25 + y \ Simplify and solve: \ 2 = 1.25 + y \ leads to \ y = 0.75 \.

4. **Combine Coordinates:** Conclude with point Q(4.5, 0.75).

By breaking down the problem into individual steps and solving for each coordinate separately, finding unknown points becomes manageable and straightforward.