# Vertices of a Square

Vertex (plural: vertices) is a place in geometry where common endpoints of two or more rays or line segments meet.

Thus, in a square, the vertex is the point where two sides meet. It is each corner of a square. Thus, each square has 4 vertices. Shown below are the 4 vertices of the □ ABCD.

## In Coordinate Plane

In coordinate geometry, a square is similar to an ordinary square with the following properties:

• All 4 sides are equal (congruent)
• Opposite sides are parallel to each other
• Diagonals bisect each other at 90°
• Diagonals are perpendicular to each other
• All 4 interior angles are right angles

In addition to the above properties, its position on the coordinate plane is also given to us. Each of the 4 vertices has known coordinates. From these coordinates, its side length, height, and some other parameters can be found.

### Finding the Diagonal Coordinates and Side Length

Suppose we are given the coordinates of any two vertices or the two vertices of a diagonal of the □ ABCD, we can find the other two unknown vertices too. The approach is based on the fact that the lengths of all the sides of a square are equal.

Since the x-coordinates are equal (x = 1) and the y-coordinates differ by a distance of 2, we can say that the length of each side of a square is 2 units.

Given, B (1, 2) = (x1, y1) and D (3, 4) =(x2, y2), then the coordinates of the other two vertices will be A = (1, 4) and C = (3, 2)

Let us now find out how we have calculated the other two unknown vertices ‘A’ and ‘C’.

If the x-coordinates of the given two vertices are equal then the coordinates of the other two vertices can be determined using the formula given below:

(x1 + y2 – y1, y1) and (x2 + y2 – y1, y2)

=> (1 + 4 – 2, 2) and (3 + 4 – 2, 4)

=> (3, 2) and (5, 4)

If the y-coordinates of the given two vertices are equal, then the coordinates of the other two vertices can be determined using the formula given below:

(x1, + y1 + x2 – x1) and (x2,  y2 + x2 – x1)

=> (1, 2 + 3 – 1) and (3, 4 + 3 – 1)

=> (1, 4) and (3, 6)

Since, ABCD is a square, coordinates (5, 4) and (3, 6) are not possible. Thus, the coordinates of A = (1, 4) and C = (3, 2). They are also the vertices of the diagonal AC.

Alternative Method

Given that the two diagonals B and A are B (1, 2) = (x1, y1) and D (3, 4) = (x2, y2), the other two vertices will be:

(x1, y2) and (x2, y1)

Now, to find the side length of □ ABCD, we will use the distance formula:

Length (AB) = √(x2 – x1)2+ (y2 – y1)2, here x1= 1, y1 = 4, x2 = 1, y2 = 2

= √(1 – 1)2 + (2 – 4)2

= 2 units

Thus, the side length of □ ABCD is 2 units

### Finding Area and Perimeter

Since, we know the side length of □ ABCD; we can find its area and perimeter using the standard formulas for calculating the area and perimeter.

As we know,

Area (A) = a2, here a = 2 units

= (2 × 2) units

= 4 units

As we know,

Perimeter (P) = 4a, here a = 2 units

= (4 × 2) units

= 8 units