Show that the points (- 3, 2), (- 5, - 5), (2, - 3) and (4, 4) are the

1.	Show that the points (- 3, 2), (- 5, - 5), (2, - 3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.
Answer» Vertices of the rhombus are: A(- 3, 2), B(- 5, - 5), C(2, -3) and D(4, 4) We know that diagonals of a rhombus bisect each other, therefore point of intersection of diagonals is: Abscissa of Mid point of AC = \(\frac{2 - 3}2\) = \(\frac{ - 1}2\) Ordinate of Mid point of AC = \(\frac{-3 + 3}2\) = \(\frac{ - 1}2\) Abscissa of Mid point of BD = \(\frac{4 - 5}2\) = \(\frac{ - 1}2\) Ordinate of Mid point of BD = \(\frac{4 - 5}2\) = \(\frac{ - 1}2\) Since the diagonals AC and BD bisect each other at O, therefore it is a rhombus. Length of diagonal AC = \(\sqrt{(2 + 3)^2 + (-3 - 2)^2}\) = \(\sqrt{25 + 25}\) = \(\sqrt{50}\) = \(5\sqrt{2}\) units Length of diagonal BD = \(\sqrt{(4 + 5)^2 + (4 + 5)^2}\) = \(\sqrt{81 + 81}\) = \(\sqrt{162}\) = \(9\sqrt{2}\) units Area of rhombus = \(\frac{1}2\times{d1}\times{d2}\) = \(\frac{1}2\times{5\sqrt2}\times{9\sqrt2}\) = 45 sq units Area of rhombus is 45 sq units

Show that the points (- 3, 2), (- 5, - 5), (2, - 3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.

Answer»

Vertices of the rhombus are: A(- 3, 2), B(- 5, - 5), C(2, -3) and D(4, 4)

We know that diagonals of a rhombus bisect each other, therefore point of intersection of diagonals is:

Abscissa of Mid point of AC = \(\frac{2 - 3}2\) = \(\frac{ - 1}2\)

Ordinate of Mid point of AC = \(\frac{-3 + 3}2\) = \(\frac{ - 1}2\)

Abscissa of Mid point of BD = \(\frac{4 - 5}2\) = \(\frac{ - 1}2\)

Ordinate of Mid point of BD = \(\frac{4 - 5}2\) = \(\frac{ - 1}2\)

Since the diagonals AC and BD bisect each other at O, therefore it is a rhombus.

Length of diagonal AC

= \(\sqrt{(2 + 3)^2 + (-3 - 2)^2}\) = \(\sqrt{25 + 25}\) = \(\sqrt{50}\) = \(5\sqrt{2}\) units

Length of diagonal BD

= \(\sqrt{(4 + 5)^2 + (4 + 5)^2}\) = \(\sqrt{81 + 81}\) = \(\sqrt{162}\) = \(9\sqrt{2}\) units

Area of rhombus = \(\frac{1}2\times{d1}\times{d2}\) = \(\frac{1}2\times{5\sqrt2}\times{9\sqrt2}\) = 45 sq units

Area of rhombus is 45 sq units