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Thread: how to show that 4 lines make a quadrilateral?

  1. #1
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    how to show that 4 lines make a quadrilateral?

    Hi folks,

    I have been given the vector equation of two lines and then asked to find the point of intersection A. Then I am given two more lines and asked to show that these lines form the other two sides of a quadrilateral. I am not sure how to do this (please help). We are then asked to find the other 3 points of intersection (B, C and D - which was no problem) and then to decide whether or not the points lie in a plane. Again no problem. I am giving this extra information in case any of it helps with the decision about the lines forming a quadrilateral.

    many thanks
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  2. #2
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    Re: how to show that 4 lines make a quadrilateral?

    if you've got 4 line segments that intersect pairwise at 4 distinct points all lying in the same plane you've got a quadrilateral.

    It sounds like you've proved what you're asking.
    Thanks from s_ingram
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    Re: how to show that 4 lines make a quadrilateral?

    Thanks romsek,

    I just wanted to be sure. The question asks for things in a specific order, so it asks me to show that the lines form a quadrilateral before asking for all the points of intersection and whether the points lie in a plane. It then goes on to ask me to provide the very things I need to establish the quad. This formulation confused me. Thanks for the clarification.
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  4. #4
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    Re: how to show that 4 lines make a quadrilateral?

    Find pairwise for products. If you get 0 for any pair, you have parallel lines. 4 nonparallel lines must intersect at 4 points. You can have up to two pairs of parallel lines and still show you get four points of intersection.
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    Re: how to show that 4 lines make a quadrilateral?

    You got the answer in your question already. Line segments that intersect pairwise at 4 distinct points all lying in the same plane make a quadrilateral. If you want more explanation you may ask at https://studydaddy.com/geometry-homework-help usually they provide great answers at tricky questions.
    Thanks from s_ingram
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    Re: how to show that 4 lines make a quadrilateral?

    Quote Originally Posted by ReseacherPh View Post
    You got the answer in your question already. Line segments that intersect pairwise at 4 distinct points all lying in the same plane make a quadrilateral. If you want more explanation you may ask at https://studydaddy.com/geometry-homework-help usually they provide great answers at tricky questions.
    This is pure SPAM.
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    Re: how to show that 4 lines make a quadrilateral?

    Careful Plato: he'll take "pure" as a compliment!!
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    Re: how to show that 4 lines make a quadrilateral?

    Quote Originally Posted by s_ingram View Post
    I have been given the vector equation of two lines and then asked to find the point of intersection A. Then I am given two more lines and asked to show that these lines form the other two sides of a quadrilateral. We are then asked to find the other 3 points of intersection (B, C and D - which was no problem) and then to decide whether or not the points lie in a plane, (Again no problem).
    Here are the relevant facts you need.
    $1~\bullet~$ any three non-collinear points determine a plane.
    $2~\bullet~$ this problem requires four points, no three of which are collinear.
    $3~\bullet~$ the problem requires a criterion for being coplanar.
    $4~\bullet~$ the problem requires an understanding plane-separation.

    Can you argue that no three of points $A,~B,~C\text{, or }D$ are collinear?
    Let’s say that is the case. If you can show that
    $\overrightarrow {AB} \cdot \left( {\overrightarrow {AC} \times \overrightarrow {AD} } \right) = 0$ then the four points are coplanar.

    At this point you need to find text material on the plane separation axioms & theorems.
    If there four coplanar points, no three of which are collinear, there are two of which in the same half-plane determined by the other two.
    There is your quadrilateral.
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