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Math Help - quadrilateral question

  1. #1
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    quadrilateral question

    number of quadrilaterals that can be made by using the vertices of a polygon of 10 sides as their vertices and having exactly two sides a common with polygon

    i have got the answer for 3 sides common with the polygon as 10
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  2. #2
    MHF Contributor Unknown008's Avatar
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    If you take only quadrilaterals with two common adjacent sides, you have 5x10 = 50 quadrilaterals. (5 possible quadrilaterals for each of the 10 pair of adjacent sides)

    Now, taking non adjacent sides, it will be a problem similar to:
    How many different pair of letters you can make from a through j, without repeat and order is not important, ie ab and ba are the same, provided two adjacent letters cannot be paired, including a and j.

    You have the sides:
    ad
    ae
    af
    ag
    ah
    be
    bf
    bg
    bh
    bi
    cf
    cg
    ch
    ci
    cj
    dg
    dh
    di
    da (but you already have ad!, so this is rejected. I won't put other repeats)
    eh
    ei
    ej
    fi
    fj
    gj

    For a total of (24 + 50) possible quadrilaterals.
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  3. #3
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    Quote Originally Posted by Unknown008 View Post
    If you take only quadrilaterals with two common adjacent sides, you have 5x10 = 50 quadrilaterals. (5 possible quadrilaterals for each of the 10 pair of adjacent sides)

    Now, taking non adjacent sides, it will be a problem similar to:
    How many different pair of letters you can make from a through j, without repeat and order is not important, ie ab and ba are the same, provided two adjacent letters cannot be paired, including a and j.

    You have the sides:
    ad
    ae
    af
    ag
    ah
    be
    bf
    bg
    bh
    bi
    cf
    cg
    ch
    ci
    cj
    dg
    dh
    di
    da (but you already have ad!, so this is rejected. I won't put other repeats)
    eh
    ei
    ej
    fi
    fj
    gj

    For a total of (24 + 50) possible quadrilaterals.
    thanks
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  4. #4
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    Lexington, MA (USA)
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    Hello, prasum!

    Number of quadrilaterals that can be made by using the vertices of a polygon
    of 10 sides as their vertices and having exactly two sides in common with polygon.

    From the available 10 vertices, choose a pair that is adjacent.
    . . There are 10 such pairs.


    For the second pair of adjacent vertices,
    . . we must not use the vertices adjacent to the first two. .**

    From the available 6 vertices, choose a pair that is adjacent.
    . . There are 5 such pairs.


    Therefore, there are: . 10 \times 5 \:=\:50 such quadrilaterals.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    **

    Label the vertices: . A,B,C,D,E,F,G,H,I,J in that order.

    Suppose we select \,AB for the first pair.

    Then we must not select C\!D or I\!J for the second pair,
    . . or we have a quadrilateral with three sides in common.

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