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