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Math Help - Area of rhombus

  1. #1
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    Question Area of rhombus

    I am having problems doing the following (please have a look at the attached figure):

    Find the area of a rhombus whose side is 15 cm long while one of the diagonals is 24 cm long. Here AD=15 cm & AC=24 cm.
    (Hint: Find OD using Pythagoras theorem)

    Thanks,

    Ron
    Attached Thumbnails Attached Thumbnails Area of rhombus-rhombus.gif  
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  2. #2
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    Hello rn5a
    Quote Originally Posted by rn5a View Post
    I am having problems doing the following (please have a look at the attached figure):

    Find the area of a rhombus whose side is 15 cm long while one of the diagonals is 24 cm long. Here AD=15 cm & AC=24 cm.
    (Hint: Find OD using Pythagoras theorem)

    Thanks,

    Ron
    Following my answer to your previous post, the other way of finding the area of a rhombus is to use the formula:
    Area of rhombus = \tfrac12 product of the lengths of the diagonals
    The reason this formula works is because the diagonals of a rhombus are perpendicular, thus dividing the rhombus into four right-angled triangles. The area of each triangle is \tfrac12 base x height; when you multiply this by 4, you'll find that you get the formula above.

    So, use the hint you're given to find OD (which is half the length of one of the diagonals), and then multiply it by the length of the other diagonal.

    Grandad
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  3. #3
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    Area of rhombus = product of the lengths of the diagonals
    The reason this formula works is because the diagonals of a rhombus are perpendicular, thus dividing the rhombus into four right-angled triangles. The area of each triangle is base x height; when you multiply this by 4, you'll find that you get the formula above.
    Sorry but I couldn't exactly understand the above which I have quoted.

    base x height x 4 = 2 base x height

    So how does the Area of rhombus = product of the lengths of the diagonals?

    Thanks,

    Ron
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  4. #4
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    Grandad explained it already:

    The reason this formula works is because the diagonals of a rhombus are perpendicular, thus dividing the rhombus into four right-angled triangles. The area of each triangle is base x height; when you multiply this by 4, you'll find that you get the formula above.
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  5. #5
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    Hello rn5a
    Quote Originally Posted by rn5a View Post
    Sorry but I couldn't exactly understand the above which I have quoted.

    base x height x 4 = 2 base x height

    So how does the Area of rhombus = product of the lengths of the diagonals?

    Thanks,

    Ron
    In your diagram, if the diagonals meet at O, then the four triangles: AOD, DOC, COB, BOA are congruent, and each is right-angled at O. The area of one of them, for example AOD, is \tfrac12AO\times OD. So the area of the rhombus is 4\times\tfrac12AO\times OD=2\times AO \times OD.

    But AO = \tfrac12 AC and OD = \tfrac12 DB. So the area of the rhombus = 2\times\tfrac12 AC \times\tfrac12 DB=\tfrac12AC\times DB=\tfrac12 the product of the diagonals.

    Grandad
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