Question

give solution of this question.

Answer 1

Answer:

Area of the triangle is 2a²√3 unit^2.

Step-by-step explanation:

If the given points are forming an equilateral triangle, distance between the sides should be same.

Given points : ( a , a ) , ( - a , - a ) , ( - √3a , √3a )

By Distance Formula : -

= > Distance between ( a , a ) and ( - a , - a )

= > $\sqrt{(a+a)^2 +(a+a)^2}$

= > $\sqrt{4a^2 + 4a^2}$

= > $\sqrt{8a^2}$

= > Distance between ( - a , - a ) and ( - √3a , √3a )

= > $\sqrt{(-a+\sqrt{3}a)^2 + (-a-\sqrt{3}a)^2}$

= > $\sqrt{a^2+3a^2-2\sqrt{3}a+a^2+3a^2+2\sqrt{3}a}$

= > $\sqrt{8a^2}$

= > Distance between ( a , a ) and ( - √3a , √3a )

= > $\sqrt{(a+\sqrt{3}a)^2 + (a-\sqrt{3}a)^2}$

= > $\sqrt{a^2+3a^2+2\sqrt{3}a+a^2+3a^2-2\sqrt{3}a}$

= > $\sqrt{8a^2}$

Since the distance between all the points is same, sides of the triangle are equal. And as we have studied, triangle with equal sides is known as equilateral triangle.

Hence, proved that the triangle formed by ( a , a ) , ( - a , - a ) and ( - √3a , √3a ) is an equilateral triangle.

From above, we have got that the distance between the points is $\sqrt{8a^2}$, it means that the length of the side of equilateral triangle is $\sqrt{8a^2}$ .

From the properties of equilateral triangle, we know that the area of equilateral triangle is $\dfrac{\sqrt{3}}{4}side^2$

Thus,

= > Area of the triange = ( √3 / 4 ) x ( $\sqrt{8a^2}$^2

= > Area of the triangle = ( √3 / 4 ) x 8a^2

= > Area of the triangle = 2a²√3 unit^2