The point P has coordinates (x, y) with respect to the origin O. By writing x = r cos and
y = r sin , or otherwise, show that, if the line OP is rotated by 60 clockwise about O,
the new y-coordinate of P is 1
2 (y −
p
3 x). What is the new y-coordinate in the case of an
anti-clockwise rotation by 60 ?
An equilateral triangle OBC has vertices at O, (1, 0) and ( 1
2 , 1
2
p
3), respectively. The point P
has coordinates (x, y). The perpendicular distance from P to the line through C and O is h1;
the perpendicular distance from P to the line through O and B is h2; and the perpendicular
distance from P to the line through B and C is h3.
Show that h1 = 1
2
y −
p
3 x
and find expressions for h2 and h3.
Show that h1 + h2 + h3 = 1
2
p
3 if and only if P lies on or in the triangle OBC.