using the concept of slope,prove that A(4,3)B(6,4) C(5,6) D(3,5) are the angular points of a square.

For a square all sides should be equal, and perpendicular to each other
So slope of AB = m₁ = (4-3)/(6-4) = 1/2
Slope of BC = m₂ = (6-4)/(5-6) = -2
Slope of CD = m₃ = (5-6)/(3-5) =1/2
Slope of DA = m₄ = (3-5)/(4-3) = -2

As AB is perpendicular to BC so m₁⨯m₂ = -1
So m₁⨯m₂ = 1/2⨯-2 = -1

Similarly you can check for other sides.

For the square AB = BC = CA = AD
So AB = √(4-3)² + (6-4)² = √5
BC = √(6-4)² + (5-6 )² = √5

Similarly check for other sides
Hence All the sides are equal and perpendicular to each other. So it is a square