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Question
The vertex of an equilateral triangle is (2, 3) and the equation of the opposite side is x + y = 2. Then the other two sides are y – 3 = `(2 +- sqrt(3)) (x - 2)`.
Options
True
False
Solution
This statement is True.
Explanation:
Let ABC be an equilateral triangle with vertex (2, 3) and the opposite side is x + y = 2 with slope –1.
Suppose slope of line AB is m.
Since each angle of equilateral triangle is 60°.
∴ Angle between AB and BC
tan 60° = `|(-1 - m)/(1 + (-1)m)|`
⇒ `sqrt(3) = |(1 + m)/(1 - m)|`
⇒ `sqrt(3) = +- ((1 + m)/(1 - m))`
Taking (+) sign,
`sqrt(3) = (1 + m)/(1 - m)`
⇒ `sqrt(3) - sqrt(3)m = 1 + m`
⇒ `sqrt(3)m + m = sqrt(3) - 1`
⇒ `m(sqrt(3) + 1) = sqrt(3) - 1`
⇒ `m = (sqrt(3) - 1)/(sqrt(3) + 1)
⇒ `m = (sqrt(3) - 1)/(sqrt(3) + 1) xx (sqrt(3) - 1)/(sqrt(3) - 1)`
⇒ `m = (3 + 1 - 2sqrt(3))/(3 - 1)`
= `2 - sqrt(3)`
Taking (–) sign,
`m = 2 + sqrt(3)`
So, the equations of other two lines are y – 3 = `(2 +- sqrt(3))(x - 2)`
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