What is a Square?
A square is a parallelogram that is equilateral and equiangular. The measure of each angle of a square is always 90°. The diagonals of a square are equal in length and they bisect each other at right angles.
Properties of a Square
- The opposite sides are parallel to each other.
- Diagonals are perpendicular bisectors of each other.
- The sides are perpendicular to each other.
- The diagonal of a square is equal to \(\sqrt {2}\) times its side.
- The diagonal of a square divides it into four congruent triangles of equal areas.
- If the side of a square is ‘a’, its perimeter is equal to 4a.
- If the side of a square is ‘a’, its area is equal to a\(^2\).
What is a Rhombus?
A rhombus is a parallelogram that is equilateral but not always equiangular. The diagonals of a rhombus are not equal but bisect each other at a right angle.
The Properties of a Rhombus
- The opposite sides are parallel to each other.
- The diagonals are perpendicular bisectors of each other.
- The sides are equal in length
- A diagonal bisects a pair of opposite angles.
- If the side of a rhombus is ‘a’, its perimeter is equal to 4a.
- If the diagonals of a rhombus are \(d_1\) and \(d_2\) respectively, its area is: \(\frac{1}{2}\times d_1 \times d_2\)
Differences between a Square and a Rhombus
Square | Rhombus |
|---|---|
All the angles of the square are equal and they are 90°. | Opposite angles of the rhombus are equal but need not be a right angle. |
Diagonals are equal and equal to \(\sqrt{2}\) times of the side of the square. | Diagonals are not of equal length in a rhombus. |
All squares are rhombuses. | All rhombuses cannot be squares, because rhombuses are not always equiangular. |
The adjacent sides of the square are perpendicular to each other. | The adjacent sides of the rhombus are not always perpendicular to each other. |
The measure of an angle of a square bisected by a diagonal is 45°. | In a rhombus, the diagonals bisect the angles, but they need not always be 45°. |
If the side of a square is ‘a’, its area is equal to a\(^2\). | If the diagonals of a rhombus are \(d_1\) and \(d_2\), respectively, its area is \(\frac{1}{2} \times d_1 \times d_2\) |
We can inscribe a square in a circle. | We can not inscribe a rhombus in a circle. |
A square is always symmetrical about four lines. | A rhombus is always symmetrical about two lines. |
Similarities between a Square and a Rhombus
Square | Rhombus |
|---|---|
A square is a parallelogram. | A rhombus is also a parallelogram. |
The measures of the sides of a square are equal. | The measures of the sides of a rhombus are equal. |
In a square, the opposite sides are parallel to each other. | In a rhombus, the opposite sides are also parallel to each other. |
The diagonals of a square are perpendicular bisectors. | The diagonals of the rhombus are also perpendicular bisectors. |
The sum of all interior and exterior angles is 360°. | The sum of all interior and exterior angles is 360°. |
If the side of a square is ‘a’, its perimeter will be ‘4a’. | If the side of the rhombus is ‘a’, its perimeter will be 4a. |
Solved Examples
Example 1: Identify whether the parallelogram ABCD is square or rhombus.
Solution:
Given that, the parallelogram ABCD is equilateral.
So, it can be a square or a rhombus.
If the diagonals are of equal length, it is a
square, otherwise it is a rhombus.
We know that the diagonals of a parallelogram bisect each other.
So, AC = AO + OC
= 5 + 5
= 10 cm
And, BD = BO + OD
= 7 + 7
= 14 cm
We can see that, AC \(\neq\) BD
Since AC \(\neq\) BD, ABCD is not a square and it is a rhombus.
Example 2: Check whether the given parallelogram ABCD with vertices A(2,0), B(7,0), C(7,5), and D(2,5) is a square or not.
Solution :
Find all the lengths of the sides of the parallelogram.
AB = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) Distance formula
= \(\sqrt{(7-2)^2+(0-0)^2}\) Substitute
= \(\sqrt{5^2}\) Evaluate square root
AB = 5
BC = \(\sqrt{(x_3-x_2)^2+(y_3-y_2)^2}\) Distance formula
= \(\sqrt{(7-7)^2+(5-0)^2}\) Substitute
= \(\sqrt{5^2}\) Evaluate square root
BC = 5
CD = \(\sqrt{(x_4-x_3)^2+(y_4-y_3)^2}\) Distance formula
= \(\sqrt{(2-7)^2+(5-5)^2}\) Substitute
= \(\sqrt{(-5)^2}\) Evaluate square root
CD = 5
DA = \(\sqrt{(x_1-x_4)^2+(y_1-y_4)^2}\) Distance formula
= \(\sqrt{(2-2)^2+(0-5)^2}\) Substitute
= \(\sqrt{(-5)^2}\) Evaluate square root
DA = 5
AC = \(\sqrt{(x_3-x_1)^2+(y_3-y_1)^2}\) Distance formula
= \(\sqrt{(7-2)^2+(5-0)^2}\) Substitute
= \(\sqrt{5^2+5^2}\) Evaluate square root
AC = \(5\sqrt{2}\)
BD = \(\sqrt{(x_4-x_2)^2+(y_4-y_2)^2}\) Distance formula
= \(\sqrt{(2-7)^2+(5-0)^2}\) Substitute
= \(\sqrt{(-5)^2+5^2}\) Evaluate square root
BD = \(5\sqrt{2}\)
Side AB = BC = CD = DA
and diagonals BD = AC
Hence, we can say that ABCD is a square.
A parallelogram is a quadrilateral whose opposite sides are parallel.
The key difference between a square and a rectangle is that all the sides of a square are equal in length, whereas in a rectangle, only the opposite sides are equal in length.
A quadrilateral is a closed figure with four sides. So, a square is a quadrilateral.
If the angles of a rhombus are right angles, we can say that the rhombus is a square.