FROM THE PERIMETER OF SQUARE FROM THE LENGTH OF DIAGONAL

Problem 1 :

Find the side length of the square.

Solution :

AB = s, AC = 6, BC = s

Using Pythagorean theorem.

(AC)² = (AB)² + (BC)²

6² = s² + s²

36 = 2s²

Dividing 2 on each sides.

36/2 = s²

18 = s²

s = √18

s = √(9 × 2)

s = 3√2

So, the sides of a square is 3√2. 

Problem 2 :

The perimeter of a square is 400 m. calculate the length of the diagonal of the square.

Solution :

Let x be the length of the diagonal of the square.

The perimeter of a square is 400 m.

Perimeter of a square = 4 × side

400 = 4 × side

Side = 400/4 = 100

AB = 100, AD = 100, BD = x

Using Pythagorean theorem.

(BD)² = (AB)² + (AD)²

x² = (100)² + (100)²

x² = 10000 + 10000

x² = 20000

x = √20000

x = √(100 × 100 × 2)

= 100√2

So, the length of the diagonal of the square is 100√2.

Problem 3 :

The perimeter of a square is 36 m. calculate the length of the diagonal of the square.

Solution :

Let x be the length of the diagonal of the square.

The perimeter of a square is 36 m.

Perimeter of a square = 4 × side

36 = 4 × side

Side = 36/4 = 9   

AB = 9, AD = 9, BD = x

Using Pythagorean theorem.

(BD)² = (AB)² + (AD)²

x² = 9² + 9²

x² = 81 + 81

x² = 162

x = √162

x = √(81 × 2)

x = √(9 × 9 × 2)

x = 9√2

So, the length of the diagonal of the square is 9√2.

Use the information marked on the figure to find the value of x.

Problem 4 :

Solution :

AB = 16, BC = 16, AC = x

Using Pythagorean theorem.

(AC)² = (AB)² + (BC)²

x² = (16)² + (16)²

x² = 256 + 256

x² = 512

x = √512

x = √(256 × 2)

x = 16√2

So, the value of x is 16√2.

Find the perimeter of quadrilateral WXYZ.

Problem 5 :

Solution ;

OY = x, OX = x, XY = 18√2

Using Pythagorean theorem.

(XY)² = (OX)² + (OY)²

(18√2)² = x² + x²

2x² = 324 × 2

2x² = 648

Dividing 2 on each sides.

2/2x² = 648/2

x² = 324

x = √(81 × 4)

x = 18

ZY = ZO + OY

43 = ZO + 18

ZO = 43 – 18

ZO = 25

OY = 18, OX = 18, XY = 18√2

WX = 25, ZO = 25, WZ = 18

So, the perimeter of quadrilateral WXYZ is,

WX + WZ + ZO + OX + XY = 25 + 18 + 25 + 18 + 18√2

= 86 + 18√2

Problem 6 :

If the perimeter of a square is 8, which is the length of the diagonal ?

A) 2√2       B) 2√3     C) 8√2        D) 4

Solution :

Let x be the length of the diagonal of the square.

The perimeter of a square is 8.

Perimeter of a square = 4 × side

8 = 4 × side

Side = 8/4 = 2  

AB = 2, AD = 2, BD = x

Using Pythagorean theorem.

(BD)² = (AB)² + (AD)²

x² = 2² + 2²

x² = 4 + 4

x² = 8

x = √8

x = √(4 × 2)

x = 2√2

So, the length of the diagonal is 2√2.

Hence, option A) is correct.