Show that the Points A(3,0), B(4,5), C(-1,4) and D(-2,-1) Are the Vertices of a Rhombus. Find Its Area. - Mathematics

Show that the points A(3,0), B(4,5), C(-1,4) and D(-2,-1) are the vertices of a rhombus. Find its area.

Solution

The given points are  A(3,0), B(4,5), C(-1,4) and D(-2,-1)

AB = sqrt((3-4)^2 + (0-5)^2 ) = sqrt((-1)^2 +(-5)^2)

= sqrt(1+25) = sqrt(26)

BC = sqrt((4+1)^2 +(5-4)^2) = sqrt((5)^2 +(1)^2)

= sqrt(25+1) = sqrt(26)

CD = sqrt((-1+2)^2 +(4+1)^2) = sqrt((1)^2 +(5)^2)

= sqrt(1+25) = sqrt(26)

AD = sqrt((3+2)^2 +(0+1)^2) = sqrt((5)^2 +(1)^2)

= sqrt(25+1) = sqrt(26)

AC = sqrt((3+1)^2 + (0-4)^2) = sqrt((4)^2+(-4)^2)

= sqrt(16+16) =4sqrt(2)

BD = sqrt((4+2)^2 +(5+1)^2 ) = sqrt((6)^2+(6)^2)

= sqrt((36+36)) = 6 sqrt(2)

∵ AB = BC =CD =AD = 6 sqrt(2) and AC ≠ BD

Therefore, the given points are the vertices of a rhombus

Area (Δ ABCD ) =1/2 xx AC xxBD

= 1/2 xx 4 sqrt(2) xx 6 sqrt(2 ) = 24  sq. units

Hence, the area of the rhombus is 24 sq. units.

Concept: Coordinate Geometry
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APPEARS IN

RS Aggarwal Secondary School Class 10 Maths
Chapter 16 Coordinate Geomentry
Q 28