Question

if two vertices of an equilateral triangle are (3,0)and(6,0),find the third vertex

Answer 1

Let ∆ABC be the equilateral triangle

Let the coordinates of vertices of ∆ABC be A (3, 0), B (6, 0) and C (x, y).

∆ABC is an equilateral triangle.

∴ AB = BC = CA
⇒ AB2 = BC2 = CA2
⇒ [(3 – 6)2 + (0 – 0)2] = [(x – 6)2 + (y – 0)2] = [(x – 3)2 + (y – 0)2]
⇒ (–3)2 = (x – 6)2 + y2 = (x – 3)2 + y2
⇒ 9 = (x – 6)2 + y2 = (x – 3)2 + y2
∴ (x – 6)2 + y2 = (x – 3)2 + y2
⇒ (x – 6)2 = (x – 3)2
⇒ x – 6 = ± (x – 3) [By taking square root on both the sides]
⇒ x – 6 = x – 3 or x – 6 = –(x – 3)
⇒ x – x = –3 + 6 or x – 6 = –x + 3
⇒ 0 = 3
(not possible)
or x + x = 3 + 6
⇒ 2x = 9

or x = 4.5
also, (x-6)^2 /+y^2=9
= (4.5-6)^2+ y^2=9
=(-1.5)^2+Y^2=9
=2.25+Y^2=9
Y^2=9-2.25
y^2=6.75
y=+-root 6.75
y=+- 2.6

Thus, coordinates of point C are (4.5, 2.6) or (4.5, – 2.6).