Question

Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer. Also find the area.
i) $(-1,-2),(1,0),(-1,2),(-3,0)$
ii) $(-3,5),(3,1),(0,3),(-1,-4)$
iii) $(4,5),(7,6),(4,3),(1,2)$

Answer 1

Distance formula $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

i)

$A(-1,-2),B(1,0),C(-1,2),D(-3,0)$

Vertices: $AB=2\sqrt{2},BC=2\sqrt{2},CD=2\sqrt{2},AD=2\sqrt{2}$

Diagonals: $AC=4,BD=4$

Since, all sides and both the diagonals are same.

ABCD is a square.

ii)

$A(-3,5),B(3,1),C(0,3),D(-1,-4)$

Vertices: $AB=\sqrt{52},BC=\sqrt{13},CD=\sqrt{50},AD=\sqrt{85}$

Diagonals: $AC=\sqrt{13},BC=\sqrt{13}$

Since, all sides are not equal to each other and both the diagonals are same.

ABCD is a quadrilateral.

iii)

$A(4,5),B(7,6),C(4,3),D(1,2)$

Vertices: $AB=\sqrt{10},BC=\sqrt{18},CD=\sqrt{10},AD=\sqrt{18}$

Since, opposite sides are equal.

ABCD is a parallelogram.