Help with geometry? Coordinate geometry proofs?

I need some help with geometry. I literally have no clue how to do it. There's two questions I need help with. Both need a proof for them.

1. Use coordinate geometry to prove that the quadrilateral OPQR with vertices O (0,0),P (9,3), Q (9,8), R (-3,4) is an isosceles trapezoid.

2.Use coordinate geometry to prove that quadrilateral PART with vertices P (-2,1), A (1,4), R (4,1), and T (1,-2) is a square.

Thanks for your help and have a wonderful day.

4 Answers

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  • Tony
    Lv 6
    7 years ago

    o.k. I'll give you "some" help, but you can do the work yourself.

    1. you have a quadrilateral with certain vertices. You wish to prove it is an isosceles trapazoid. What do you know about an isosceles trapazoid???

    Well, you know that:

    1. the bases are parallel

    2. the sides are equal

    3. the "base" is longer than the "top"

    What do you have to work with?? only the co-ordinate points.

    Well, it 'pears to ME that you can use the distance formula to find the length of the sides and show that they are equal. You can then find the slope of the base and top to show they are parallel. Then you can find the length of the base and top to show they are UNEQUAL and thus you do not have a parallelogram.

    problem #2. What are the characteristics of a square. Well, if I recall correctly, it is 4 sides equal and one right angle. Thus, you can use the distance formula to show all the sides are equal. THEN, you can use the distance formula to show one diagonal is the sum of the squares of two sides, thus proving the included angle is a right angle.

    Don't make this stuff harder than it is.

    always,

    tony

  • Anonymous
    7 years ago

    1) Trapezoids have one set of parallel sides, and isosceles trapezoids have a pair of congruent sides. Plot OPQR on a coordinate plane to help you visualize it. Then, find the slopes of OP, PQ, QR, and RO using the slope formula (m=y2-y1/x2-x1). Parallel lines have the same slopes and it should work out that there is one set of parallel lines thus proving that it is a trapezoid. Now use the distance formula (d=sqrt((y2-y1)^2+(X2-X1)^2)) to find the lengths of all the legs. You should end up with two congruent distances thus proving the trapezoid to be isosceles.

    2) You can prove that all sides are congruent to each other and that both the diagonals are congruent to each other (the sides will not be equal to the diagonals). You must perform the distance formula on all four sides and you must perform the distance formula on the two diagonals.

    The second possible solution is to prove that adjacent sides have negative reciprocal slopes, hence perpendicular and opposite sides have equal slopes. You must also prove that the slopes of the diagonals have negative reciprocal slopes. You use the slope formula on all four sides and you use the slope formula on the two diagonals, MT and ES.

    Good Luck !!! :)

  • Erika
    Lv 4
    3 years ago

    Trapezoid Proofs

  • maget
    Lv 4
    3 years ago

    enable the kite ABCD have A on (0, 0) and B on (2a, 0) as you reported on x-axis. whence AB = 2a = CB and DA = DC as in line with definition of kite. Now if different 2 factors are C(c, ok) and D(h, d) then slope of AC is okay / c and slope of BD is d / (h -- 2a) For AC to be perpendicular to BD, we ought to continuously coach kd / c(h -- 2a) = --a million OR kd = (2a -- h)c Now as AB = 2a = CB and DA = DC as in line with definition of kite, CB^2 = 4a^2 and DA^2 = DC^2 Giving (2a -- c)^2 + ok^2 = 4a^2 and h^2 + d^2 = (h -- c)^2 + (d -- ok)^2 giving c^2 + ok^2 = 4ac and h^2 + d^2 = h^2 + d^2 -- 2hc + c^2 -- 2dk + ok^2 OR 2kd = c^2 + ok^2 -- 2hc = 4ac -- 2hc = 2(2a -- h)c Whence kd = (2a -- h)c Proving AC perpendicular to BD.

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