Question

HELPP! ABCD is a square. The equation of the diagonal AC is 3x + 2y - 7 = 0. What is the equation of the diagonal BD, if the coordinates of vertex B are (3,5)?

Answer 1

2x - 3y + 9 = 0 which is the first option

Explanation:

We make use of the fact that the diagonals of a square bisect each other at right angles

In other words, diagonal BD is perpendicular to diagonal AC

Equation of AC = 3x + 2y - 7 = 0

Convert this standard form to slope-intercept form

3x + 2y - 7 = 0

• Add 7 to both sides
  ==> 3x + 2y - 7 + 7 = 0 + 7
  ==> 3x + 2y = 7
  
• Subtract 3x from both sides:
  ==> 3x - 3x + 2y = - 3x + 7
  ==> 2y = -3x + 7
  
• Divide by 2 both sides:
  ==> 2y/2 = (-3/2)x +7/2
  ==> y = (-3/2)x + 7/2

• The slope of AC = -3/2

• A perpendicular line will have a slope which is the negative of the reciprocal of the slope of this line

• reciprocal of -3/2 = -2/3
  Negative of this value = -(-2/3) = 2/3

• So equation of BD is
  y = (2/3)x + b where b is the y-intercept
  
• To find b, plug in the coordinates of point B(3, 5) into the above equation:
  5 = 2/3 x 3 + b
  5 = 2 + b
  b = 3
  
• So the equation of the line in slope intercept form is
  y = (2/3)x + 3
  
• Convert this to standard form:
  Multiply throughout by 3
  ==> 3y = 2x + 9
  
• Move 3y to the right
  ==> 0 = 2x - 3y + 9
• Rewrite as
  2x - 3y + 9 = 0

Answer
2x - 3y + 9 = 0 which is the first option