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Math Help - Surd problem

  1. #1
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    Surd problem

    hi all,

    this is a problem that requires the answer expressed as a surd. (calculator not to be used)

    The sides of a rectangle are in the ratio 2:3. The diagonal is of length 26cm. Find the perimeter.

    The answer given in the answer section is 20\sqrt13

    26^2 = 626

    sides are therefore

    \sqrt{\frac{2}{5} * 626} \mbox{ and} \sqrt{\frac{3}{5} * 626}

    and perimeter

    2\sqrt{\frac{1352}{5}} + 2\sqrt{\frac{2028}{5}}

    using prime factors of 1352 and 2028

    26\sqrt{\frac{8}{5}} + 52\sqrt{\frac{3}{5}}

    using a calculator the answer i calculated is not equal to the textbooks answer
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  2. #2
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by sammy28 View Post
    hi all,

    this is a problem that requires the answer expressed as a surd. (calculator not to be used)

    The sides of a rectangle are in the ratio 2:3. The diagonal is of length 26cm. Find the perimeter.

    The answer given in the answer section is 20\sqrt13

    26^2 = 626

    sides are therefore

    \sqrt{\frac{2}{5} * 626} \mbox{ and} \sqrt{\frac{3}{5} * 626}

    and perimeter

    2\sqrt{\frac{1352}{5}} + 2\sqrt{\frac{2028}{5}}

    using prime factors of 1352 and 2028

    26\sqrt{\frac{8}{5}} + 52\sqrt{\frac{3}{5}}

    using a calculator the answer i calculated is not equal to the textbooks answer
    First of all 26^2 =676

    I dont know what you are trying to do

    Lets say the sides are 2x & 3x

    Hence
    Diagonal = \sqrt{4x^2+9x^2} = 26

    x\sqrt{13} = 26

    Divide both sides by squareroot(13)

    x = 2\sqrt{13}

    Perimeter = 2(2x + 3x) = 10 x = 20\sqrt{13}
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  3. #3
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    excellent thanks adarsh.

    i didnt think about using algebra, i was just taking the ratio as a total 676=5/5 and expected the result to be the same. I need to figure out where i went wrong
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  4. #4
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    i played with this abit more im still confused as to why the above not work but using adarsh algebra

    26 = \sqrt{(\frac{2}{5}x)^2 + (\frac{3}{5}x)^2}

    26 = \sqrt{\frac{4}{25}x^2 + \frac{9}{25}x^2}

    26=\sqrt{\frac{13}{25}x^2} \equiv x\sqrt{\frac{13}{25}}

    x = 26 \div \sqrt{\frac{13}{25}} \equiv \frac{26 * 5}{\sqrt{13}} \equiv \frac{13 * 10}{13^\frac{1}{2}} \equiv 10\sqrt{13}

    perimeter 2 x sides

    2 \mbox{x} 10\sqrt{13} \equiv 20\sqrt{13}
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  5. #5
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by sammy28 View Post
    i played with this abit more im still confused as to why the above not work but using adarsh algebra

    26 = \sqrt{(\frac{2}{5}x)^2 + (\frac{3}{5}x)^2}

    26 = \sqrt{\frac{4}{25}x^2 + \frac{9}{25}x^2}

    26=\sqrt{\frac{13}{25}x^2} \equiv x\sqrt{\frac{13}{25}}

    x = 26 \div \sqrt{\frac{13}{25}} \equiv \frac{26 * 5}{\sqrt{13}} \equiv \frac{13 * 10}{13^\frac{1}{2}} \equiv 10\sqrt{13}

    perimeter 2 x sides

    2 \mbox{x} 10\sqrt{13} \equiv 20\sqrt{13}
    I think you got it But just as an assurance I will do it your way

    Lets consider the sides to be 3x/5 and 2x/5

    Now we follow the steps (same as you did here)

    26 = \sqrt{(\frac{2}{5}x)^2 + (\frac{3}{5}x)^2} ....this thing came from Pythagoras theorem

    26 = \sqrt{\frac{4}{25}x^2 + \frac{9}{25}x^2}

    26=\sqrt{\frac{13}{25}x^2} \equiv x\sqrt{\frac{13}{25}}

    x = 26 \div \sqrt{\frac{13}{25}} \equiv \frac{26 * 5}{\sqrt{13}} \equiv \frac{13 * 10}{13^\frac{1}{2}} \equiv 10\sqrt{13}

    Perimeter is the sum of lengths of all sides

    Thus what we basically did was

    (3x/5 + 2x/ 5) + (3x/5 + 2x/5) = x + x = 2x =Answer
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  6. #6
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    thank you for explaining it so thoroughly, my problem solving skill is not as good as it should be. I need to think before jumping into questions
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