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Why is the diagonal of a square longer than its side?

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In mathematics, the proportions, measurements, dimensions, forms, shapes, and angles of different objects around us are studied under the branch called geometry. Geometry is one of the oldest branches of mathematics and has a very significant application and impact on our daily lives. We are surrounded by innumerable objects in our lives. All these objects possess a certain shape, take up significant space, can be used to store a particular amount of things in them, and can be placed as per different positions. All these factors fall under the scope of geometry. Shapes can be categorized a two- dimensional and three-dimensional. 

Two-Dimensional Shapes

The sub-branch of geometry that deals with two- dimensional shapes is called plane geometry. It is concerned with such shapes and figures as can be drawn on a piece of paper. Two- dimensional shapes, as the name suggests are made up of only two proportions: length and breadth, and can be drawn on a cartesian plane.

Square

A square is a quadrilateral whose all four sides are of equal length and all four angles are of equal measure as well. The study of square falls under the scope of plane geometry only since it is a 2D shape. The adjacent sides of a square intersect at right angles. The following figure depicts a square with sides AB = CD and AC = BD, and ∠A = ∠B = ∠C = ∠D = 90°.

Diagonals of a Square

There are two diagonals in a square, which connect the opposite vertices to each other while bisecting each other at 90°. The following figure shows a square ABCD with both of its diagonals, AB and CD. Both AB and CD bisect each other at point O such that ∠AOB = ∠BOC = ∠COD = ∠DOA = 90°.

Why is the diagonal of a square longer than its side?

Solution:

The diagonal of a square divides it into two right triangles. If only one diagonal were to be constructed, say diagonal BD, then it would divide the whole square into two congruent right triangles, â–³BDC and â–³ABD. This is shown in the following image:

Let the length of side of the square be a units.

Upon the first glance, it is obvious that the diagonal BD is the hypotenuse while DC and BC are the perpendicular and base respectively. We know that the hypotenuse id the longest side in a right triangle, hence, the diagonal of a square (hypotenuse of â–³BDC) is greater than its side (BC/ DC).

In right- triangle BDC, using Pythagoras Theorem, we have:

BD2 = BC2 + CD2

⇒ BD2 = a2 + a2

⇒  BD2 = 2a2

⇒ BD = √2a

Hence, Diagonal of a square > Side of the square.

Sample Problems

Question 1. Find the diagonal of a square whose side is √2 cm.

Solution:

Diagonal of a square = √2 × side

Here, Side = √2 cm

⇒ D = √2 × √2 cm

Thus, diagonal of the square = 2 cm.

Question 2. Find the perimeter of a square whose diagonal is 10√2 cm.

Solution:

Diagonal of a square = √2 × side

⇒ Side = Diagonal/ √2 

= 10√2/ √2

= 10 cm

Perimeter of the square = 4(10) cm

= 40 cm.

Question 3. Find the area of a square whose diagonal is 10√2 cm.

Solution:

Diagonal of a square = √2 × side

⇒ Side = Diagonal/ √2

= 10√2/ √2

= 10 cm

Area of the square = 102 sq. cm

= 100 sq. cm.

Question 4. Find the diagonal of a square if its side is 6√2 cm.

Solution:

Diagonal of a square = √2 × side

Here, Side = 6√2 cm

⇒ D = √2 × 6√2 cm

Thus, diagonal of the square = 12 cm.

Question 5. Find the area and perimeter of a square if its diagonal is 3√2 cm.

Solution:

Diagonal of a square = √2 × side

⇒ Side = Diagonal/ √2

= 3√2/ √2

= 3 cm

Perimeter of the square = 4(3) cm

= 12 cm.

Area of the square = 32 sq. cm.

= 9 sq. cm. 


Last Updated : 29 Dec, 2021
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