Question

a square and an equilateral triangle have equal perimeter if the diagonol of the square is 6√2 cm then the area of the triangle is a ( tell in cm2 or cm 3)​

Answer 1

Answer:

Area = 16√3 cm²

Step-by-step explanation:

Given :-

A square and an equilateral triangle have equal perimeters.

The diagonal of the square is 6√2 cm

To find :-

The area of the triangle

Solution :-

Let the side of a square be a cm

Given that

The diagonal of the square = 6√2 cm

We know that

The diagonal of a square of a units is a√2 units

Therefore, 6√2 = a√2 cm

=> a√2 = 6√2

=> a = 6 cm

Therefore, The side of the square = 6 cm

We know that

Perimeter of a square of a units is 4a units

Perimeter of the given square

=> Perimeter = 4×6 cm

=> Perimeter = 24 cm

Therefore, Perimeter of the square

= 24 cm

According to the given problem

Perimeter of a square = Perimeter of an equilateral triangle

=> Perimeter of an equilateral triangle

= 24 cm

We know that

Perimeter of an equilateral triangle is 3×length of its side .

=> 3×side of the triangle = 24 cm

=> Side of the triangle = 24/3

=> Side of the triangle = 8 cm

Therefore, Side of the equilateral triangle is 8 cm

We know that

Area of an equilateral triangle

= (√3/4) a² sq.units

Where, a is the side of the triangle

Area of the given triangle

=> A = (√3/4)×8² cm²

=> A = (√3/4)×8×8

=> A = (√3/4)×64

=> A = √3 × 16

=> A = 16√3 cm²

or

=> A = 16×1.732

=> A = 27.712 cm² ( approximately)

Therefore , Area = 16√3 cm² or

27.71 cm²

Answer:-

Area of the equilateral triangle is

16√3 cm² or 27.71 cm²

Used formulae:-

→ The diagonal of a square of a units

= a√2 units

→ Perimeter of a square of a units

= 4a units

→ Perimeter of an equilateral triangle

= 3×length of its side .

→Area of an equilateral triangle

= (√3/4) a² sq.units

Where, a is the side of the triangle

→ √3 = 1.732...

Answer 2

Given :-

A square and an equilateral triangle have equal perimeter if the diagonal of the square is 6√2 cm

To Find :-

Area of triangle

Solution :-

First we need to find the side of square

In ΔABC

AB² + BC² = AC²

s² + s² = (6√2)²

2s² = 36 × 2

2s² = 72

s² = 72/2

s² = 36

s = √36

s = 6

Now

We know that

Perimeter of square = 4 × side

P = 4 × 6

P = 24 cm

Now

Perimeter of equilateral triangle = Perimeter of square

24 cm = Perimeter of equilateral triangle

24 cm  = 3 × side

24/3 = side

8 cm = side

Now

Area of equilateral triangle = √3/4 × a²

Area = √3/4 × (8)²

Area = √3/4 × 64

Area = √3 × 16

Area = 16√3 cm²