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# Show that the Points A (1, −2), B (3, 6), C (5, 10) And D (3, 2) Are the Vertices of a Parallelogram. - CBSE Class 10 - Mathematics

#### Question

Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.

#### Solution

The distance d between two points (x_1,y_1) and (x_2, y_2) is given by the formula

d = sqrt((x_1 - x_2)^2 +(y_1 - y_2)^2)

In a parallelogram the opposite sides are equal in length.

Here the four points are A(1, −2), B(3, 6), C(5, 10) and D(3, 2).

Let us check the length of the opposite sides of the quadrilateral that is formed by these points.

AB = sqrt((1 - 3)^2 + (2 - 6))

=  sqrt((-2)^2 + (-8)^2)

= sqrt(4 + 64)

AB = sqrt(68)

CD = sqrt((5 - 3)^2 + (10 - 2)^2)

= sqrt((2)^2 + (8)^2)

= sqrt(4 + 64)

CD = sqrt(68)

We have one pair of opposite sides equal.

Now, let us check the other pair of opposite sides.

BC = sqrt((3 - 5)^2 + (6 - 10)^2)

= sqrt((-2)^2 + (-4)^2)

=sqrt(4 + 16)

AD = sqrt20

The other pair of opposite sides is also equal. So, the quadrilateral formed by these four points is definitely a parallelogram.

Hence we have proved that the quadrilateral formed by the given four points is a parallelogram

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Solution Show that the Points A (1, −2), B (3, 6), C (5, 10) And D (3, 2) Are the Vertices of a Parallelogram. Concept: Distance Formula.
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