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Math Help - Equilateral Triangle

  1. #1
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    Equilateral Triangle

    PQR is an equilateral triangle. The point U is the mid-point of PR. Points T and S divide QP and QR in the ratio 1:2. The point of intersection of PS, RT and QU is X. If the area of QSX is 1 sqaure unit, what is the area in square units of PQR.


    the middle point is X
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  2. #2
    MHF Contributor red_dog's Avatar
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    We have A[QSX]=1\Rightarrow A[SRX]=2\Rightarrow A[QPX]=3

    Let ST\cap QU=M.

    QM=\frac{1}{3}QU, \ ST=\frac{1}{3}PR, \ MX=\frac{1}{3}XU

    \frac{MX}{XU}=\frac{1}{3}\Rightarrow\frac{MX}{MU}=  \frac{1}{4}

    Then MX=\frac{1}{4}MU=\frac{1}{4}(QU-QM)=\frac{1}{6}QU

    QX=QM+MX=\frac{1}{2}QU\Rightarrow A[XUR]=A[QXR]=3

    Then A[QUR]=6\Rightarrow A[PQR]=12
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  3. #3
    Senior Member pacman's Avatar
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    Hint: Use Area-Ratio Method or maybe 2-pole problem

    reddog is very fast
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by jgv115 View Post
    PQR is an equilateral triangle. The point U is the mid-point of PR. Points T and S divide QP and QR in the ratio 1:2. The point of intersection of PS, RT and QU is X. If the area of QSX is 1 sqaure unit, what is the area in square units of PQR.


    the middle point is X
    QX = (1/2)QU

    QR =(1/3) perp height from S on to QX

    therefore area QXS is (1/6) area of QUR.

    Hence area of PQR=12.

    CB
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  5. #5
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    Quote Originally Posted by red_dog View Post
    We have A[QSX]=1\Rightarrow A[SRX]=2\Rightarrow A[QPX]=3

    Let ST\cap QU=M.

    QM=\frac{1}{3}QU, \ ST=\frac{1}{3}PR, \ MX=\frac{1}{3}XU

    \frac{MX}{XU}=\frac{1}{3}\Rightarrow\frac{MX}{MU}=  \frac{1}{4}

    Then MX=\frac{1}{4}MU=\frac{1}{4}(QU-QM)=\frac{1}{6}QU

    QX=QM+MX=\frac{1}{2}QU\Rightarrow A[XUR]=A[QXR]=3

    Then A[QUR]=6\Rightarrow A[PQR]=12
    mm i only understand parts of that.

    <br />
\frac{MX}{XU}=\frac{1}{3}\Rightarrow\frac{MX}{MU}=  \frac{1}{4}<br />


    this part.

    I'm assuming your made up "M" is if you draw a straight line from T to S right in the middle? Correct me if I'm wrong.

    How do you get that?
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