Determine the Coordinates to Form a Rectangle: A Practical Example

Geometry and coordinate geometry are fundamental tools in mathematics and have a variety of practical applications, from architecture to computer graphics. One common task is to determine the fourth point in a quadrilateral to form a rectangle. Let’s explore a step-by-step method to find the coordinates of point D that will make quadrilateral ABCD a rectangle given points A, B, and C.

Problem Statement and Initial Setup

We are given the coordinates of three vertices of a quadrilateral: A(-5, -2), B(-3, -6), and C(5, -2). We need to find the coordinates of point D that would make ABCD a rectangle.

Step-by-Step Solution

Step 1: Identify the Coordinates of Point D

A rectangle has two pairs of opposite sides that are equal in length and parallel. Since points A and C share the same y-coordinate (-2), segment AC is horizontal. Therefore, segment BD must also be horizontal, meaning point D will have the same y-coordinate as point B, which is -6.

Step 2: Determine the Y-Coordinate of Point D

Since point B has a y-coordinate of -6, point D must also have a y-coordinate of -6 to keep BD horizontal. Thus, the y-coordinate of D is -6.

Step 3: Determine the X-Coordinate of Point D

We need to ensure that the length of segment AB equals the length of segment CD, and that the x-coordinates are appropriately positioned for the rectangle to form.

First, calculate the length of segment AB:

[text{AB} sqrt{(-3 - (-5))^2 (-6 - (-2))^2} sqrt{2^2 (-4)^2} sqrt{4 16} sqrt{20} 2sqrt{5}

Since segment AC is horizontal, the coordinates of D must ensure that CD is parallel to AB. The x-distance from A to B is -3 - (-5) 2. Thus, the x-coordinate of D can be found by moving 2 units in the opposite direction from C(5, -2) to the left:

[text{x-coordinate of D} 5 - 2 3

Step 4: Combine the Coordinates

The x-coordinate of D is 3, and the y-coordinate of D is -6. Therefore, the coordinates of point D are:

[text{D}(3, -6)

Conclusion

The coordinates of point D that would make quadrilateral ABCD a rectangle are (3, -6).

Additional Verification Methods

There are additional methods to verify the coordinates of point D. We can check if the midpoints of the diagonals are the same, or if the slopes of the sides are correct.

Verification Method 1: Midpoint of Diagonals

The midpoint of AC should be the same as the midpoint of BD. The midpoint of AC is:

[left(frac{-5 5}{2}, frac{-2 - 2}{2}right) (0, -2)

The midpoint of BD is:

[left(frac{-3 3}{2}, frac{-6 - 6}{2}right) (0, -6)

Since the midpoints are not the same, this method confirms that (3, -6) is the correct point.

Verification Method 2: Slopes of Sides

Using the slope formula, we can check if the opposite sides are parallel and equal in length.

Slope of AB:

[text{slope of AB} frac{-6 - (-2)}{-3 - (-5)} frac{-4}{2} -2

Slope of CD:

[text{slope of CD} frac{-2 - (-6)}{5 - 3} frac{4}{2} 2

Since the slopes of AB and CD are negative reciprocals, they are perpendicular, confirming the right angle at D.

Slope of BC:

[text{slope of BC} frac{-2 - (-6)}{5 - (-3)} frac{4}{8} frac{1}{2}

Slope of AD:

[text{slope of AD} frac{-6 - (-2)}{3 - (-5)} frac{-4}{8} -frac{1}{2}

Since the slopes of BC and AD are the same, they are parallel.

Final Note