maths proving question?

Slope of the line joining (6, 3) and (9, 5)
= (3 - 5)/(6 - 9)
= (-2)/(-3)
= 2/3

Slope of the line joining (5, 7) and (9, 1)
= (7 - 1)/(5 - 9)
= 6/(-4)
= -3/2

[Slope of the line joining (6, 3) and (9, 5)] × [Slope of the line joining (5, 7) and (9, 1)]
= (2/3) × (-3/2)
= -1

Hence, the line joining (6,3) and (9,5) is perpendicular to the line joining (5,7) and (9,1).

The line passing through (5,7) and perpendicular to the line joining (6,3) and (9,5) and is (9-6)x+(5-3)y=(9-6)5+(5-3)7 which simplifies to 3x+2y=29.
The line passing through (9,1) and perpendicular to the line joining (6,3) and (9,5) and is (9-6)x+(5-3)y=(9-6)9+(5-3)1 which simplifies to 3x+2y=29.
This is the same line, so the line perpendicular to the line joining (6,3) and (9,5) and passing through (5,7), also passes through (9,1).
Hence the line joining (6,3) and (9,5) is perpendicular to the line joining (5,7) and (9,1)

m1 = 2/3
m2 = - 6/4 = - 3/2
m1 x m2 = - 1 thus lines are perpendicular

How to get the equation of the line that passes through A (6 ; 3) B (9 ; 5) ?

How to get the equation of the line that passes through C (5 ; 7) D (9 ; 1) ?


The typical equation of a line is: y = mx + b → where m: slope and where b: y-intercept


For the line (AB), let's calculate m₁

m₁ = (yB - yA) / (xB - xA) = (5 - 3) / (9 - 6) = 2/3


For the line (CD), let's calculate m₂

m₂ = (yD - yC) / (xD - xC) = (1 - 7) / (9 - 5) = - 6/4 = - 3/2


Two lines are perpendicular if the product of their slope is - 1.

= m₁ * m₂

= (2/3) * (- 3/2)

= - 1 → you can say that (AB) is perpendicular to the line (CD)