Midpoint Coordinates Explanation

Understanding the concept of midpoint coordinates is crucial in coordinate geometry. The midpoint of a line segment connecting two points is the point exactly halfway between them. This problem involves finding the coordinates of B given the midpoint of line AB and the coordinates of A.

Finding Point B Coordinates

To determine the coordinates of B when given the midpoint and coordinates of A, we utilize the standard midpoint formula. This formula helps us relate the coordinates of the two endpoints to the coordinates of their midpoint.

Let the coordinates of point A be $$(x_1, y_1)$$, point B be $$(x_2, y_2)$$, and the midpoint M be $$(x_m, y_m)$$.

The midpoint formula is precisely defined as:

• For the x-coordinate: $$x_m = \frac{x_1 + x_2}{2}$$
• For the y-coordinate: $$y_m = \frac{y_1 + y_2}{2}$$

Given Coordinates and Unknowns

From the problem statement, we are provided with the following coordinates:

• The Midpoint M $$(x_m, y_m)$$ is (14, 4).
• The coordinates of Point A $$(x_1, y_1)$$ are (-4, -22).
• The coordinates of Point B $$(x_2, y_2)$$ are unknown. Let's denote them as (x, y).

Our goal is to find the values of x and y for point B.

Calculating x-coordinate of B

We will apply the midpoint formula specifically for the x-coordinate using the given values:

$$x_m = \frac{x_1 + x_2}{2}$$

Substitute the known values into the formula:

$$14 = \frac{-4 + x}{2}$$

To solve for x, first multiply both sides of the equation by 2:

$$14 \times 2 = -4 + x$$

$$28 = -4 + x$$

Now, add 4 to both sides to isolate x:

$$28 + 4 = x$$

$$x = 32$$

Thus, the x-coordinate of point B is 32.

Calculating y-coordinate of B

Similarly, we will use the midpoint formula for the y-coordinate:

$$y_m = \frac{y_1 + y_2}{2}$$

Substitute the known y-values into the formula:

$$4 = \frac{-22 + y}{2}$$

Multiply both sides of the equation by 2:

$$4 \times 2 = -22 + y$$

$$8 = -22 + y$$

Now, add 22 to both sides to isolate y:

$$8 + 22 = y$$

$$y = 30$$

Thus, the y-coordinate of point B is 30.

Final Coordinates of B

Based on our calculations, the coordinates of point B are (32, 30). This process demonstrates how to effectively use the midpoint formula to find an unknown endpoint when the other endpoint and the midpoint are known.