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Question
The area of the figure formed by joining the mid-points of the adjacent sides of a rhombus with diagonals 16 cm and 12 cm is
Options
28 cm2
48 cm2
96 cm2
24 cm2
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Solution
Given: Rhombus with diagonals measuring 16cm and 12 cm.
To find: Area of the figure formed by lines joining the midpoints of the adjacent sides.
Calculation: We know that, ‘Area of a rhombus is half the product of their diagonals’.
H and F are the midpoints of AD and BC respectively.
`AH = 1/2 AD `
`BE = 1/2 BC`
Now ABCD is a parallelogram which means
`AD = BC `
`1/2AD = 1/2BC`
AH = BF ……..(1)
Also , AD || BC
AH || BF ……(2)
From 1 and 2 we get that ABFH is a parallelogram.
Since Parallelogram FHAB and ΔFHE are on the base FH and between the same parallels HF and AB.
∴` ar (Δ FHE ) = 1/2 ar ( "||"^(gm) FHAB )` ……(3)
Similarly ,
`ar (ΔFHG) = 1/2ar ("||"^(gm) FHDC)` ……(4)
Adding 3 and 4 we get,
`ar (Δ FHE ) + ar (ΔFHG) = 1/2 ar ("||"^(gm) FHAB)+1/2ar("||"^(gm)FHDC)`
`ar (EFGH) = 1/2 (ar("||"^(gm) FHAB ) + ar ("||"^(gm) FHDC))`
`ar (EFGH) = 1/2 (ar("||"^(gm) ABCD))`
`ar (EFGH) = 1/2 (1/2 (16xx12))`
`ar (EFGH) = 1/4 (16 xx 12)`
`ar (EFGH) = 48 cm^2`
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