Question

If the length of diagonal of a square is 4-√2 cm, find it's length, perimeter and area.​

Answer 1

Explanation:

1) the length of the side is:

`(4-√(2) )/(√(2))=2√(2)-1.`

2) the required perimeter is:

`P=4(2√(2) -1)=8√(2) -4.`

3) the required area is:

`A=(2√(2) -1)^2=9-4√(2).`

Answer 2

Side length = (2√2 - 1) units

Perimeter = (8√2 - 4) units

Area = (9 - 4√2) units²

Explanation:

Properties of a square:

• It is a quadrilateral
• The opposite sides are parallel
• All four sides are equal in length
• All interior angles measure 90°

The diagonal of a square creates two right triangles, with the diagonal being the hypotenuse, and the sides of the square being the 2 legs.

Let x = side length of the square

Using Pythagoras' Theorem: a² + b² = c²

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

⇒ x² + x² = (4 - √2)²

⇒ 2x² = 18 - 8√2

⇒ x² = 9 - 4√2

⇒ x = ±(2√2 - 1)

⇒ x = 2√2 - 1 only (as distance is positive)

Perimeter of a square = 4x

= 4(2√2 - 1)

= 8√2 - 4

Area of a square = x²

= (2√2 - 1)²

= 9 - 4√2