Question:
Prove that the points (a,a) (-a,-a) (-aroot3,aroot3) are the vertices of an equilateral triangle.
Answer:
Let ABC is a triangle where A (-a, -a), B(a, a) and C(-a√3, a√3)
If the triangle ABC is equilateral then
AB = BC = CA
Now, AB = √[{a - (-a)}2 + {a - (-a)}2 ]
=> AB = √[{a + a}2 + {a + a}2 ]
=> AB = √(4a2 + 4a2 )
=> AB = √(8a2 )
=> AB = 2a√2
Again,
BC = √[{a - (-a√3)}2 + {a - (a√3)}2 ]
=> BC = √[{a + a√3}2 + {a - a√3}2 ]
=> BC = √[a2 (1 + √3)2 + a2 (1 - √3)2 ]
=> BC = √[a2 (1 + 3 + 2√3) + a2 (1 + 3 - 2√3) ]
=> BC = √[4a2 + 2a2 √3 + 4a2 - 2a2 √3) ]
=> BC = √(8a2 )
=> BC = 2a√2
And
CA = √[{-a√3 - (-a)}2 + {a√3 - (-a)}2 ]
=> CA = √[{a - a√3}2 + {a + a√3}2 ]
=> CA = √[a2 (1 - √3)2 + a2 (1 + √3)2 ]
=> CA = √[a2 (1 + 3 - 2√3) + a2 (1 + 3 + 2√3) ]
=> CA = √[4a2 - 2a2 √3 + 4a2 + 2a2 √3) ]
=> CA = √(8a2 )
=> CA = 2a√2
So, AB= BC = CA
Hence, ABC is an equilateral triangle where A (-a, -a), B(a, a) and C(-a√3, a√3)
Not what you are looking for? Go ahead and submit the question, we will get back to you.
