Question

the side of an equilateral triangle is decreased by 2cm , its area is decreased by 11√3 cm². what is the height of the triangle?
A.. 3√3 cm
B.. 4√3 cm
C. 5√3 cm
D. 6√3 cm​

Answer 1

Answer:

Option D

Step-by-step explanation:

Given :-

The side of an equilateral triangle is decreased by 2cm , its area is decreased by 11√3 cm².

To find :-

What is the height of the triangle?

Solution :-

Let the side of an equilateral triangle be x cm

Area of an equilateral triangle

= (√3/4)x² cm²

Given that

The side of an equilateral triangle is decreased by 2cm

Then, the side of the new triangle

= (x-2) cm

Then Area of the new equilateral triangle

=> (√3/4)(x-2)² cm²

According to the given problem

Area of the equilateral triangle is decreased by 11√3 cm²

So, The New area = (√3/4)x²-11√3 cm²

=> (√3/4)(x-2)² = (√3/4)x²-11√3

=> (√3/4)(x-2)² - (√3/4)x² = -11√3

=> (√3/4)[(x-2)²-x²] = -11√3

=> (√3/4)[x²-4x+4-x²] = -11√3

=> (√3/4)(-4x+4) = -11√3

=> -4x+4 = (-11√3)×(4/√3)

=> -4x+4 = -11×4

=> -4x+4 = -44

=> -4x = -44-4

=> -4x = -48

=> 4x = 48

=> x = 48/4

=> x = 12 cm

Side of the equilateral triangle = 12 cm

We know that

Height of the equilateral triangle

= (√3/2) x cm

=>Height = (√3/2)×12 cm

=> Height = (√3×6 ) cm

=> Height = 6 √3 cm

Therefore, Height = 6 √3 cm

Answer:-

The height of the equilateral triangle is

6 √3 cm

Used formulae:-

→ Area of an equilateral triangle

= (√3/4)a² sq.units

→ Height of an equilateral triangle

= (√3/2) a units

• a = Side of the equilateral triangle

Answer 2

Solution :-

Let us assume that, each side of equilateral triangle is x cm .

since we know that,

• Area of an equilateral triangle with each side as a is = (√3/4)a²

A/q,

→ (√3/4)x² - (√3/4)(x - 2)² = 11√3

→ (√3/4)[x² - (x - 2)²] = 11√3

→ x² - x² + 4x - 4 = 44

→ 4(x - 1) = 44

→ x - 1 = 11

→ x = 11 + 1

→ x = 12 cm .

therefore,

→ Height of Equaliteral ∆ = (√3/2) * side

→ Required height = (√3/2) * 12

→ Required height = 6√3 cm (Ans.)

Learn more :-

In the figure ∠ MNP = 90°, ∠ MQN = 90°, , MQ = 12 , QP = 3 then find NQ .

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