# Show that the Points a (1, 0), B (5, 3), C (2, 7) and D (−2, 4) Are the Vertices of a Parallelogram. - Mathematics

Show that the points A (1, 0), B (5, 3), C (2, 7) and D (−2, 4) are the vertices of a parallelogram.

#### Solution

Let A (1, 0); B (5, 3); C (2, 7) and D (-2, 4) 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) = ((1 + 2)/2, (0 + 7)/2)

= (3/2, 7/2)

Similarly mid-point of diagonal BD is,

R(x,y) = ((5 - 2)/2, (3 + 4)/2)

= (3/2, 7/2)

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

Hence ABCD is a parallelogram.

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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.3 | Q 34 | Page 29