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Math Help - Quadrilateral Problem

  1. #1
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    Quadrilateral Problem

    In quadrilateral ABCD, diagonals AC and BD intersect at P. E, F, G, H are the centroids of triangles ABP, BCP, CDP, and ADP respectively. If the area of the quad ABCD is 18, find the area of the quad EFGH. The centroid of a triangle is the point of intersection of the medians of the triangle.

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  2. #2
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    Quote Originally Posted by spred View Post
    In quadrilateral ABCD, diagonals AC and BD intersect at P. E, F, G, H are the centroids of triangles ABP, BCP, CDP, and ADP respectively. If the area of the quad ABCD is 18, find the area of the quad EFGH. The centroid of a triangle is the point of intersection of the medians of the triangle.

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    1. Draw a sketch of the quadrilateral. (The difficulty here is to find a quadrilateral which has no further properties)

    2. You can prove that

    \overline{HE}\ \parallel\ \overline{GF}\ \parallel\ \overline{DB}

    and

    \overline{EF}\ \parallel\ \overline{HG}\ \parallel\ \overline{AC}

    That means the quadrilateral EFGH must be a parallelogram.

    3. The quadrilateral EFGH consists of 4 smaller parallelograms which are part of the 4 triangles which form the quadrilateral ABCD. As an example I've taken the triangle BCP and the corresponding parallelogram KFLP. Use proportions to show that the area of KFLP is \frac29 of the triangle BCP.

    4. You'll get this result in each of the 4 triangles. Therefore the area of the parallelogram EFGH must be \frac29 of the area of the quadrilateral ABCD.
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