Q52 of 59 Page 178

State whether the statements are true or false.

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 .

Let ABC be an equilateral triangle with vertex (2, 3)

and equation of opposite side is x + y = 2


We know that, all angle of an equilateral triangle is of 60°


So, θ = 60°


Let the slope of line AB is m


and slope of the given equation x + y = 2 is



∴ m2 = -1


We know that,



Putting the values of m1 and m2 in above eq., we get






⇒ (1 – m)√3 = 1 + m or (1 – m)(-√3) = – 1 – m


⇒ √3 - √3 m = 1 + m or -√3 + √3 m = – 1 – m


⇒ √3 – 1 = m + √3m or -√3 + 1 = - m - √3m


⇒ √3 – 1 = m(1 + √3) or -(√3 – 1) = -m(√3 + 1)


…(i) or …(ii)


Now, multiply and divide eq. (i) by √3 + 1, we get






⇒ m = 2 + √3


Now, multiply and divide eq. (i) by √3 – 1 , we get






⇒ m = 2 - √3


So, the slope of line AB is 2 ± √3


So, the equations of other two lines joining the point (2, 3) are


y – 3 = 2 ± √3 (x – 2)


Hence, the given statement is TRUE


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