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Math Help - Inscribed square in triangle

  1. #1
    Senior Member I-Think's Avatar
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    Inscribed square in triangle

    Good night. I have a geometry problem that's been plaguing me for years. Any hints you all can give would be greatly appreciated.

    PQR is a right-angled triangle with PR=3cm and QR=4cm. The square STVU is inscribed in the triangle PQR. What is the length, in cm, of the side of the square?

    All your help is much appreciated.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Hello I-Think
    Quote Originally Posted by I-Think View Post
    Good night. I have a geometry problem that's been plaguing me for years. Any hints you all can give would be greatly appreciated.

    PQR is a right-angled triangle with PR=3cm and QR=4cm. The square STVU is inscribed in the triangle PQR. What is the length, in cm, of the side of the square?

    All your help is much appreciated.
    All the triangles are similar: 3-4-5.

    If TU = x cm, then ST=UV = x cm. So from \triangle PTS, PT = \tfrac34x cm.

    And from \triangle UVQ, UQ = \tfrac43x cm.

    So, since PQ = 5 cm:

    \tfrac34x + x + \tfrac43x = 5

    \Rightarrow 9x + 12 + 16x = 60

    \Rightarrow x = \frac{60}{37}

    Grandad
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  3. #3
    Senior Member I-Think's Avatar
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    Thanks

    Merci beaucoup. To think it had such a simple solution.
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