Show that points A(-4, -7), B(-1, 2), C(8, 5) and D(5, -4) are vertices of a rhombus ABCD

Show that points A(-4, -7), B(-1, 2), C(8, 5) and D(5, -4) are vertices of a  rhombus ABCD

Answer:-

The given points are A(–4, –7), B(–1, 2), C(8, 5) and D(5, –4). 

Distance between two points [= sqrt{((x_2 – x_1)^2 + (y_2 – y_1)^2)}]

According to the distance formula,

[AB =sqrt{([-1-(-4)]^2 +[2-(-7)]^2}]

[∴ AB = sqrt{(3^2+9^2)}]

[∴ AB =sqrt{(9+81)} ]

[∴ AB = sqrt{90}                      …(1)]

[BC =sqrt{([8-(-1)]^2+(5-2)^2)}]

[∴ BC =sqrt{(9^2+3^2)}]

[∴ BC = sqrt{(81+9)}]

[∴ BC = sqrt90                     …(2)]

[CD = sqrt{((5-8)^2 +(-4-5)^2)}]

[∴ CD =sqrt((-3)^2 +(-9)^2)]

[ ∴ CD =sqrt{(9+81)}]

[∴ CD = sqrt90           ……. (3)]

[AD = sqrt{([5-(-4)]^2+ [-4-(-7)]^2)}]

[∴ AD =sqrt{(9^2+3^2)}]

[∴ AD =sqrt{(81+9)}]

[∴ AD = sqrt90                      ……. (4)]

From (1), (2), (3), and (4)

AB = BC = CD = AD

Thus, all sides are equal.

In a quadrilateral, if all the sides are equal, then it is a rhombus.

Chapter 5. Co-ordinate Geometry – Practice Set 5.1 (Page 107)

PREVIOUS


Q 6


NEXT