If two vertices of an equilateral triangle are (0, 0) and (0, 2 √3). Find the third vertices.
Answers
Answer:
The third vertex is (0, 2√3)
Step-by-step explanation:
Let the third vertex be (x,y)
Since it is an equilateral triangle,
=> all the sides will be equal
=> distance between the vertices are equal
We know that distance between (x₁,y₁) and (x₂,y₂) = √(x₂-x₁)² + (y₂-y₁)² (Distance formula)
Distance between (0, 0) and (0, 2 √3) = √(0-0)² + (2 √3-0)²
= √12 --- (i)
Distance between (0, 2 √3) and (x,y) = √(x-0)² + (y-2 √3)²
= √ x² + y² + 12 -2y√3 ---(ii)
Distance between (x,y) and (0, 0) = √(0-x)² + (0-y)²
= √ x² + y² ---(iii)
Equating equations (ii) and (ii)
√ x² + y² + 12 -2y√3 = √ x² + y²
Squaring both sides,
x² + y² + 12 -2y√3 = x² + y²
As x² + y² is common on both the sides, we can cancel the term x² + y²
=> 12 -2y√3 = 0
=> 12 = 2y√3
=> 6 = y√3
=> 2√3 = y
Equating equations (i) and (iii)
√ x² + y² = √12
Squaring both sides,
x² + y² = 12
Substituting the value of y in the above equation
=> x² + ( 2√3)² = 12
=> x² + 12 = 12
=> x² = 0
=> x = 0
Therefore, the third vertex is (0, 2√3)