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Math Help - Geometry-related proofs

  1. #1
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    Geometry-related proofs

    Not using math more advanced that the 10th Grade (thus excluding the dot product) proof that:
    a)The diagonals of a rhombus (a four-sided polygon in which every side has the same length) are perpendicular;
    b)That connecting the medium points of consecutive sides of a quadrilateral will yield a parallelogram;
    c)That if (x1;y1);(x2;y2); (x3;y3) and (x4;y4) are part of the same line then: (y2-y1)/(x2-x1)=(y3-y4)/(x3-x4).
    Please help me.
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  2. #2
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    Geometry Proofs

    Hi grizzlyjoker -

    Quote Originally Posted by grizzlyjoker View Post
    Not using math more advanced that the 10th Grade (thus excluding the dot product) proof that:
    a)The diagonals of a rhombus (a four-sided polygon in which every side has the same length) are perpendicular;
    b)That connecting the medium points of consecutive sides of a quadrilateral will yield a parallelogram;
    c)That if (x1;y1);(x2;y2); (x3;y3) and (x4;y4) are part of the same line then: (y2-y1)/(x2-x1)=(y3-y4)/(x3-x4).
    Please help me.
    (a) Prove that the diagonals divide the rhombus into 4 identical (congruent) triangles. Then use the fact that two equal angles that add up to 180 degrees (where the diagonals meet) must each be 90 degrees.

    (b) In addition to joining the four median points of the sides of the quadrilateral, join the two diagonals as well. Then use similar triangles to prove that the opposite sides of the inner quadrilateral are parallel in pairs to the two diagonals.

    (c) Use the fact that the gradient of the line joining (x_1,y_1) to (x_2, y_2) is \frac{y_2-y_1}{x_2-x_1}; repeat for the other two points, and then say that the gradients are equal.

    Can you do them now?

    Grandad
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  3. #3
    MHF Contributor
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    Geometry Proofs

    Hi grizzlyjoker -

    Quote Originally Posted by grizzlyjoker View Post
    Not using math more advanced that the 10th Grade (thus excluding the dot product) proof that:
    a)The diagonals of a rhombus (a four-sided polygon in which every side has the same length) are perpendicular;
    b)That connecting the medium points of consecutive sides of a quadrilateral will yield a parallelogram;
    c)That if (x1;y1);(x2;y2); (x3;y3) and (x4;y4) are part of the same line then: (y2-y1)/(x2-x1)=(y3-y4)/(x3-x4).
    Please help me.
    (a) Prove that the diagonals divide the rhombus into 4 identical (congruent) triangles. Then use the fact that two equal angles that add up to 180 degrees (where the diagonals meet) must each be 90 degrees.

    (b) In addition to joining the four median points of the sides of the quadrilateral, join the two diagonals as well. Then use similar triangles to prove that the opposite sides of the inner quadrilateral are parallel in pairs to the two diagonals.

    (c) Use the fact that the gradient of the line joining (x_1,y_1) to (x_2, y_2) is \frac{y_2-y_1}{x_2-x_1}; repeat for the other two points, and then say that the gradients are equal.

    Can you do them now?

    Grandad
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