Question

A(3,5)and C(7,9) are the opposite vertices of a rhombus ABCD.find the equation of diagonal BD.

Answer 1

Answer:

y = 12 - x  

Step-by-step 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.