Answer:
y = 12 - x
Explanation:
The diagonals of a rhombus bisect each other at right angles,
Thus, segment BD is a line perpendicular to AC and passing through the midpoint of AC.
1. Find the midpoint of AC
The midpoint of two points is half-way between their x- and y-values.
For the x-coordinate,
(x₂ + x₁)/2 = (7 + 3)/2 = 10/2 = 5
For the y-coordinate,
(y₂ + y₁)/2 = (9 + 5)/2 = 14/2 = 7
The coordinates of the midpoint are (5,7).
2. Calculate the equation of the diagonal BD
(a) Slope of AC
m₁ = Δy/Δx = (y₂ -y₁)/(x₂ - x₁) = (9 - 5)/(7 - 3) = 4/4 = 1
(b) Slope of BD
The slope m₂ of the perpendicular line BD must be the negative reciprocal of the slope of AC.
m₁ = -1/m₁ = -1
(c) Calculate the y-intercept of BD
The general equation for a straight line is
y = mx + b
Insert point (5,7).
7 = -1×5 + b
7 = -5 + b
b = 12
The y-intercept is (0,12).
(d) Write the equation for the line
y = -x + 12 or y = 12 - x
Points B and D can be any two points on the line that are equidistant from Point O, as shown in the Figure below.