Question

Show that the points A(–3, 2), B(–5, –5), C(2, –3) and D(4, 4) are the vertices of a rhombus. Find the area of this rhombus.

HINT: Area of a rhombus = `1/2` × (product of its diagonals).

Show that A(–3, 2), B(–5, –5), C(2, –3) and D(4, 4) are the vertices of a rhombus.

Answer 1

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 rhombus, all the sides are equal in length. And the area ‘A’ of a rhombus is given as

A = 1/2(Product of both diagonals)

Here the four points are A(−3,2), B(−5,−5), C(2,−3) and D(4,4)

First, let us check if all the four sides are equal.

`AB = sqrt((-3+5)^2 + (2 + 5)^2)`

`=sqrt((2)^2 + (7)^2)`

`=sqrt(49 + 4)`

`AB=sqrt(53)`

`BC =sqrt((-5-2)^2 + (-5+3)^2)`

`= sqrt((-7)^2 + (-2)^2)`

`=sqrt(49 + 4)`

`BC = sqrt(53)`

`CD = sqrt((2- 4)^2 + (-3 - 4)^2)`

`sqrt((-2)^2 + (-7)^2)`

`= sqrt(4 + 49)`

`CD = sqrt(53)`

`AD = sqrt((-3-4)^2 + (2 - 4)^2)`

`= sqrt((-7)^2 + (-2)^2)`

`= sqrt(49 + 4)`

`AD = sqrt53`

Here, we see that all the sides are equal, so it has to be a rhombus.

Hence we have proved that the quadrilateral formed by the given four vertices is a rhombus.

Now let us find out the lengths of the diagonals of the rhombus.

`AC = sqrt((-3-2)^2 + (2 + 3))`

`= sqrt((-5)^2 + (5)^2)`

`= sqrt(25 + 25)`

`= sqrt(50)`

`AC = 5sqrt2`

`BD = sqrt((-5-4)^2 + (-5-4)^2)`

`= sqrt((-9)^2 + (-9)^2)`

`= sqrt(81 + 81)`

`= sqrt162`

`BD = 9sqrt2`

Now using these values in the formula for the area of a rhombus we have,

`A = ((5sqrt2)(9sqrt2))/2`

`= ((5)(9)(2))/2`

A = 45

Thus the area of the given rhombus is 45 square units

Answer 2

The given points are A(-3, 2), B(-5,-5), C(2,-3) and D(4,4).

`AB = sqrt((-5+3)^2 +(-5-2)^2) = sqrt((-2)^2 +(-7) ^2) = sqrt(4+49) = sqrt(53)   units`

`BC = sqrt((2+5)^2 +(-3+5)^2 )= sqrt((7)^2 +(2)^2) = sqrt(4+49)= sqrt(53)  units`

`CD = sqrt((4-2)^2 +(4+3)^2 )= sqrt((2)^2 +(7)^2) = sqrt(4+49)= sqrt(53)  units`

`DA = sqrt((4+3)^2 +(4-2)^2 )= sqrt((7)^2 +(2)^2) = sqrt(4+49)= sqrt(53)  units`

Therefore `AB =BC=CD=DA= sqrt(53)  units`

Also, `AC =- sqrt((2+3)^2 +(-3-2)^2) = sqrt((5)^2 +(-5)^2 ) = sqrt(25+25) = sqrt(50) = sqrt(25xx2) = 5 sqrt(2)  units`

`BD = sqrt((4+5)^2 +(4+5)^2) = sqrt((9)^2 +(9)^2) = sqrt(81+81) = sqrt(162) = sqrt(81 xx 2) = 9 sqrt(2)  units`

Thus, diagonal AC is not equal to diagonal BD.

Therefore ABCD is a quadrilateral with equal sides and unequal diagonals

Hence, ABCD a rhombus
Area of a rhombus `= 1/2 xx `(product of diagonals)

`= 1/2 xx (5 sqrt(2) )xx (9 sqrt(2) )`

`(45(2))/2`

= 45 square units.