Finding the Coordinates of Vertices in a Square

Given a square ABCD with two of its coordinates as A (3, -1) and B (3, 5), how do we determine the coordinates of the other two vertices? This problem requires a step-by-step approach to understanding the properties and relationships within a square. In this article, we will explore the solution using both graphical and algebraic methods.

Graphical Approach

First, let's draw a simple graph with the given points. Since points A and B share the same x-coordinate (3), and their y-coordinates are -1 and 5, the line segment AB is vertical. This vertical line segment will help us determine the coordinates of the other two vertices, C and D, which will be either:

C and D lie in the first and fourth quadrants. C and D lie in the second and third quadrants.

Given that the side length of the square is 6 (the difference in y-coordinates of A and B), we can plot the coordinates of C and D accordingly:

C (9, 5) D (9, -1) C (-3, 5) D (-3, -1)

These points can be verified by drawing the square on graph paper.

Algebraic Approach

Using equations to solve for the coordinates of the vertices is also a valid approach. Since ABCD is a square, and AB is a vertical segment, we can use the properties of squares to find the coordinates of C and D.

1. Point C:

Since point A is below point B, we can extend the vertical side upward by the same length as the side AB. Therefore, the coordinates of point C are (3, 4), as its y-coordinate is 5 units above the y-coordinate of A (-1).

2. Point D:

To find the coordinates of point D, we extend the horizontal side of AB to the left by the same length. Point D will have the same y-coordinate as B (5) and an x-coordinate that is 2 units to the left of the x-coordinate of B (3). Therefore, the coordinates of point D are (1, 5).

Mathematical Derivation

Let's use the distance formula to verify the coordinates. Since AB BC CD DA, we can use the distance formula to find the coordinates of C and D.

Let the coordinates of C be (h, 5) and D be (h, -1).

Then, the distance AB can be calculated as:

AB 6 (since the difference in y-coordinates of A and B is 6).

Using the distance formula for BC and CD:

BC CD sqrt{(h-3)^2 (5-5)^2} sqrt{(h-3)^2}

CD AB 6

Hence, ( (h-3)^2 36 ),

Hence, ( h-3 pm 6 ),

( h 9 ) or ( h -3 )

Therefore, the coordinates of point C are (9, 5) or (-3, 5) and the coordinates of point D are (9, -1) or (-3, -1).

Conclusion

In summary, the coordinates of the vertices of the square ABCD are (C 9, 5), (D 9, -1), or (C -3, 5), (D -3, -1).