Question

An equilateral triangle has an area of $64\sqrt{3}$ $\text{cm}^2$. if each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased?

Answer 1

28sqrt(3)

Explanation:

The area of an equilateral triangle is given by √3/4 side^2.

So

64√3 = √3 /4 side^2

256 = side^2

16 cm = side of original equilateral triangle.

Decreasing each side by 4 cm produces an area of :

√3/ 4 * 12^2 = 36√3 cm^2

So the area decrease is 64√3 - 36√3 = 28√3 cm^2

Answer 2

Let

ABC------> an equilateral triangle

x-------> length side of the triangle

we know that

if ABC is an equilateral triangle

then

AB=BC=AC------> x

and

angle A=angle B=angle C=60 degrees

Applying the law of sines

Area of triangle=(1/2)*a*b*sin C

in this problem

a=b=x cm

C=60 degrees

Area=64/(√3) cm²-------> 36.95 cm²

A=(1/2)*a*b*sin C------> (1/2)*x*x*sin 60°----> (1/2)*x²*(√3)/2

64/(√3)=(1/2)*x²*(√3)/2-----> 64*4=√3*√3*x²-----> 256=3*x²

x²=256/3------> x=16/√3 cm----> x=(16√3)/3 cm

if each side of the triangle is decreased by 4 cm

the new side of triangle (x1) is equal to

x1=[(16√3)/3]-4-----> x1=[16√3-12]/3 cm

the new area (A1) decreased is equal to

A1=(1/2)*x1²*(√3)/2-----> A1=(√3/4)*[(16√3-12)/3]²----> A1=(√3/36)*[(16√3-12)]²

A1=(√3/36)*[(16√3)²-2*(16√3)*12+12²)]----> A1=(√3/36)*[(768)-(384√3)+144)]

A1=(√3/36)*[(912)-(384√3)]-----> A1=11.879 cm²

therefore

the answer is

the area decreased is 11.879 cm²