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Math Help - triangle length with cosines

  1. #1
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    triangle length with cosines

    Hi guys just need some help.

    being given this and havent got a clue.

    if you could work it out and then explain that would be great

    Points P+Q are on opposite sides of a straight tunnel. A third point R forms a triangle PQR. S+T are established along the sides PR and QR repectively such that

    PR= 312m
    SR= 116m
    ST= 275m
    QR= 423m
    RT= 287m

    I need to calculate the length of the tunnel assuming that all points are in the same place.

    Thanks PPL

    James
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, jaynlisa08!

    Points P and Q are on opposite sides of a straight tunnel.
    A third point R forms a triangle PQR.
    S and T are on sides PR and QR repectively such that :

    PR= 312m, SR= 116m, ST= 275m, QR= 423m, RT= 287m

    Find the length of PQ.
    Code:
                  R
                  o
                 *  *
                *     *
               *        *
            S o * * * * * o T
             *              *
            *                 *
           *                    *
        P o * * * * * * * * * * * o Q
    In \Delta PRT use the Law of Cosines:

    . . \cos R \:=\:\frac{RS^2 + RT^2 - ST^2}{2(RS)(RT)} \;=\;\frac{116^2 + 287^2 - 275^2}{2(116)(287)} \;=\;\frac{2525}{8323}


    In \Delta PRQ, use the Law of Cosines:

    . . PQ^2 \;=\;PR^2 + QR^2 - 2(PR)(QR)\cos R

    . . PQ^2\;=\;312^2 + 423^2 - 2(312)(423)\,\frac{2525}{8323} \;= \;198,089.3162


    Therefore: . PQ \;\approx\;442.94 m.

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