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# Triangles : The lengths of the diagonals of a rhombus are 24 cm and 32 cm. Calculate the length of the altitude of the rhombus.

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The lengths of the diagonals of a rhombus are 24 cm and 32 cm. Calculate the length of the altitude of the rhombus.

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The diagonals of a rhombus bisect each-other at right angles,with the side of the rhombus being the hypotenuse . Half the lengths of the diagonals AO = 12 cm , BO= 16 cm. and the hypotenuse(AB) in right triangle ABO in  rhombus ABCD.

Using Pyhagoras Theorem for the right-angled triangle ABO,

h² = 12² + 16² = 400.

∴ The length of the side of the rhombus is 20 cm.

area of a rhombus = ½×d1×d2

are of a rhombus= base × height

∴ comparing both formulas

½×24×32 = 20×h    (base is side of the rhombus)

∴ h= 19.2 cm.

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2. Let ABCD be a rhombus with AC and BD as its diagonals.
We know that the diagonals of a rhombus bisect each other at right angles.
Let O be the intersecting point of both the diagonals.
Let AC=24cm and BD=32cm
OA=AC/2
OA= 24/2=12cm
OB=BD/2
OB=32/2=16cm
In rt.ΔAOB by Pythagoras theorem we have
AB²=OA²+OB²
=(12)²+(16)²
=144+256
=400
AB=20cm
Hence, each side of the rhombus is of length 20cm
Area of rhombus=1/2*AC*BD
=1/2*24*32
=12*32
Area of rhombus=base*altitude
12*32=20*h
19.2cm=h

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3. The diagonals of a rhombus bisect each-other at right angles,with the side of the rhombus being the hypotenuse . Half the lengths of the diagonals AO = 12 cm , BO= 16 cm. and the hypotenuse(AB) in right triangle ABO in rhombus ABCD.

Using Pyhagoras Theorem for the right-angled triangle ABO,

h² = 12² + 16² = 400.

∴ The length of the side of the rhombus is 20 cm.

area of a rhombus = ½×d1×d2

are of a rhombus= base × height

∴ comparing both formulas

½×24×32 = 20×h (base is side of the rhombus)

∴ h= 19.2 cm.

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