# Prove that the Points (4, 5) (7, 6), (6, 3) (3, 2) Are the Vertices of a Parallelogram. is It a Rectangle. - Mathematics

Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.

#### Solution

Let A (4, 5); B (7, 6); C (6, 3) and  D (3, 2) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a parallelogram.

We should proceed with the fact that if the diagonals of a quadrilateral bisect each other than the quadrilateral is a parallelogram.

Now to find the mid-point P(x,y) of two points A(x_1, y_1) and B(x_2,y_2) we use section formula as,

P(x,y) = ((x_1 + x_2)/2, (y_1 + y_2)/2)

So the mid-point of the diagonal AC is,

Q(x,y) = ((4 + 6)/2, (5 + 3)/2)

= (5,4)

Therefore the mid-points of the diagonals are coinciding and thus diagonal bisects each other.

Hence ABCD is a parallelogram.

Now to check if ABCD is a rectangle, we should check the diagonal length.

AC = sqrt((6 - 4)^2 + (3 - 5)^2)

= sqrt(4 + )4

= 2sqrt2

Similarly,

BD = sqrt((7 - 3)^2 + (6 - 2)^2)

= sqrt(16 + 16)

= 4sqrt2

Diagonals are of different lengths.

Hence ABCD is not a rectangle.

Concept: Coordinate Geometry
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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.3 | Q 15 | Page 29