Abcd is a parallelogram, with vertices labelled anticlockwise, such that a and c are the points (-1,5) and (5,1) respectively. a) given that bd is parallel tot he line with equation y+5x=2, find the equation of bd. b) given that bc is perpendicular to ac, find: i) the equation of bc ii) the coordinates of b iii) the coordinates of d

A) BD and AC meet at a point which is the midpoint of AC,
the mid point of AC is ((-1+5)/2, (5+1)/2) = (2,3)
Line BD is also parallel to a line with the equation y= -5x +2
But parallel lines have equal slope, thus the slope of BD is -5
Therefore, since BD passes through (2,3) and has a slope of -5 , its equation will be;
(y-3)/(x-2)=-5
 y-3 = -5(x-2)
y-3 = -5x + 10
y= -5x+13

b) BC is perpendicular to AC, 
the product of slopes of two perpendicular lines is -1
The slope of AC is (1-5)/(5--1)= -4/6 that is -2/3
Therefore the gradient of BC will be 3/2
Equation of BC;
  (y-1)/(x-5) = 3/2
2(y-1) = 3(x-5)
2y - 2 = 3x - 15
   2y= 3x -13

ii) the coordinates of B
Line BC and line BD meet at point B, 
Therefore, to get the coordinate of B, we solve the equation of BC and BD simultaneously
 y + 5x =13
2y - 3x = -13 ( by eliminating y)

= 2y + 10x = 26
  2y   - 3x   = -13 (subtracting the two equations)

         13 x = 39
x = 3 and y = -5(3) +13
y= -2
thus the coordinates of B is (3,-2)

iii) the coordinates of D, 
The mid point of line BD is  (2,3) and the coordinate of B is (3,-2)
thus (x+3)/2 =2  and (y +-2)/2 = 3, solving for x and y;
therefore x = 1 and y= 8
hence the coordinate of D is (1,8)

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