prove that area of a rhombus is half the area of its diagonals

Asked by ayush kumar | 3rd Mar, 2011, 01:54: AM

Expert Answer:

Dear Student,
Area of rhombus ABCD = area of triangle ABD + area of triangle CBD
Triangles ABD and CBD are congruent by SSS
 Area of rhombus ABCD = 2×(Area of triangle ABD)
 AE is perpendicular to DB because the diagonals of a rhombus are perpendicular bisectors of each other.
Area of triangle ABD = DB×AE/2 because a triangle's area is one-half the product of a side and the altitude drawn to that side.
 Area of rhombus ABCD = 2×(Area of triangle ABD)
So area of rhombus ABCD = 2×(DB×AE/2) = DB×AE
 AE = AC/2 because the diagonals of a rhombus are perpendicular bisectors of each other.
 So area of rhombus DB×AE = DB×(AC/2) = DB×AC/2
Hence area of a rhombus is half the area of its diagonals
Regards
Team Topperlearning

Answered by  | 2nd Mar, 2011, 10:54: PM

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