The points ( a , a ) , ( – a , − a ) and ( – √ 3 a , √ 3 a ) are the vertices of a/an _____ triangle. Options: A . &nbsp scalene B . &nbsp right angled C . &nbsp isosceles right angled D . &nbsp equilateral

Answer: Option D : D Let's name the points as A ( a , a ) , B ( − a , − a ) and C ( – √ 3 a , √ 3 a ) . By distance formula, distance between points ( x 1 , y 1 ) and ( x 2 , y 2 ) = √ ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 B C = √ ( − √ 3 a + a ) 2 + ( √ 3 a + a ) 2 = √ a 2 + ( √ 3 a ) 2 − 2 × a × √ 3 a + √ 3 a ) 2 + a 2 + 2 × a × √ 3 a B C = √ 8 a 2 = 2 √ 2 a units Similarly A B = √ ( a + a ) 2 + ( a + a ) 2 = √ 8 a 2 = 2 √ 2 a units A C = √ ( − √ 3 a − a ) 2 + ( √ 3 a − a ) 2 = √ 8 a 2 = 2 √ 2 a units We get, A B = B C = A C = 2 √ 2 a units ∴ △ ABC is an equilateral triangle .

The points (a,a),(–a,−a) and (

Question

The points $(a, a), (- a, - a)$ and $(- \sqrt{3} a, \sqrt{3} a)$ are the vertices of a/an _____ triangle.

Options:

A . scalene

B . right angled

C . isosceles right angled

D . equilateral

Answer: Option D
:
D

Let's name the points as $A (a, a), B (- a, - a)$ and $C (- \sqrt{3} a, \sqrt{3} a)$ .

By distance formula, distance between points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$
$= \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$
$B C = \sqrt{(- \sqrt{3} a + a)^{2} + (\sqrt{3} a + a)^{2}} = \sqrt{a^{2} + (\sqrt{3} a)^{2} - 2 \times a \times \sqrt{3} a + \sqrt{3} a)^{2} + a^{2} + 2 \times a \times \sqrt{3} a} B C = \sqrt{8 a^{2}} = 2 \sqrt{2} a units Similarly A B = \sqrt{(a + a)^{2} + (a + a)^{2}} = \sqrt{8 a^{2}} = 2 \sqrt{2} a units A C = \sqrt{(- \sqrt{3} a - a)^{2} + (\sqrt{3} a - a)^{2}} = \sqrt{8 a^{2}} = 2 \sqrt{2} a units$