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Math Help - Different shapes with equal area

  1. #1
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    Different shapes with equal area

    Hello, I would be grateful for any help with regard to formulas for the following:

    "A piece of string is cut into two pieces of different lengths. These are then used to form two different shapes with equal areas. Find the measurements of the shapes that satisfy the conditions shown as follows.

    1. String: 90cm; Shapes: square and rectangle
    2. String: 44cm; Shapes: rectangle and a right triangle
    3. String: 52cm; Shapes: parallelogram and square"
    Thanks in advance,

    Kim
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  2. #2
    Senior Member nikhil's Avatar
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    Lightbulb My thinking

    Let me solve one else will be done similarly
    let us divide thread in a and b
    such that a+b=90cm
    area of square with side a=(a^2)/16
    Now we may calculate area of rectangle.
    Lea the side of rectangle be in c:d then
    2x(c+d)=b therefore
    x=b/2(c+d)
    therefore length of sides will be
    (cb)/2(c+d) and (db)/2(c+d)
    therefore area of rectangle
    = (cdb^2)/4(c+d)^2
    equating the areas
    a^2/16=(cdb^2)/4(c+d)^2
    a^2/4 =(cdb^2)/(c+d)^2
    a/2 =[[(cd)^1/2]b]/(c+d)
    a/2 =[[(cd)^1/2](90-a)]/(c+d)
    now put any value of c and d and get desired value of a.
    e.g if c=4 and d=1 you will get
    a=40 and b=50 you will get different answers for different value of c and d

    this solution is a general solution. This happend because a number of rectangles are possible for a given parameter.
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  3. #3
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    Thank you, Nikhil, though I'm afraid I still haven't grasped it. Just finding it all a little confuzzling. Are you (or anyone else) able to provide a clearer solution? I would really appreciate any help, feeling a little like throwing the text book out the window atm.

    The given solutions are as follows: (Just not sure how to get there)

    Problem Number 1: square: 10cm by 10cm; rectangle: 20cm by 5cm

    Problem Number 2: rectangle: 6cm by 4cm; right triangle: base 6cm, height 8cm

    Problem Number 3: parallelogram: base 9cm, height 4cm; square: 6cm by 6cm

    Thanks again,

    Kim
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  4. #4
    Senior Member nikhil's Avatar
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    Smile hmmmmm....

    Your answer of 1) is :
    square 10cm by 10 cm
    rectangle 20 cm by 5cm
    well if you see co-incidently I have given this solution.
    In my example I took ratio of rectangle sides 1:4 and got
    a=40 where a is perimeter of square so one side will be 40/4=10 cm therefore area =10 cm by 10 cm
    b=50 where b is perimeter of rectangle since sides are in 1:4 therefore sides of different length will be 20cm and 5 cm therefore area will be 20cm by 5 cm

    but answer will vary if ratio is changed like if it is taken 1:9.
    So there are many solutions.

    hope this explaination helped you
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  5. #5
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    Thank you, nikhil

    Thank you for further explanation, nikhil. I will use your example and practise with the other similar problems in the text book.

    Thanks again for taking the time.

    Kim
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