Results 1 to 7 of 7

Math Help - What are the coordinates of A

  1. #1
    rcs
    rcs is offline
    Senior Member rcs's Avatar
    Joined
    Jul 2010
    From
    iligan city. Philippines
    Posts
    455
    Thanks
    2

    What are the coordinates of A

    A trapezoid ABCD has an area of 24 sq. units. There of its coordinates are
    B(1,8) , C(2,3), and D(-6,3) . Find the coordinates of A.

    can anybody help me on this problem above please.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,829
    Thanks
    123

    Re: What are the coordinates of A

    Quote Originally Posted by rcs View Post
    A trapezoid ABCD has an area of 24 sq. units. There of its coordinates are
    B(1,8) , C(2,3), and D(-6,3) . Find the coordinates of A.

    can anybody help me on this problem above please.
    1. This problem has no unique solution, because with the given conditions you can draw 6 different trapezoids.

    2. See attachment: There are 2 trapezoids in red, 2 trapezoids in blue, 2 trapezoids in green.

    3. Take the red trapezoids (that's the simplest case!):

    Point A must be situated on a parallel to the x-axis with the equation y = 8.

    The area of a trapezoid is calculated by : A_t = \frac{|\overline{AB}|+|\overline{CD}|}{2} \cdot h

    |\overline{CD}| = 8, h = 8

    So you have to solve for |\overline{AB}|:

    24 = \frac{|\overline{AB}|+8}{2} \cdot 5

    which yields |\overline{AB}|= 1.6

    So you get 2 different points A: A_1(-0.6, 8) or A_2(2.6, 8)

    4. The next 4 trapezoids are much more difficult to determine.
    Attached Thumbnails Attached Thumbnails What are the coordinates of A-vieletrapeze.png  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    rcs
    rcs is offline
    Senior Member rcs's Avatar
    Joined
    Jul 2010
    From
    iligan city. Philippines
    Posts
    455
    Thanks
    2

    Re: What are the coordinates of A

    Quote Originally Posted by earboth View Post
    1. This problem has no unique solution, because with the given conditions you can draw 6 different trapezoids.

    2. See attachment: There are 2 trapezoids in red, 2 trapezoids in blue, 2 trapezoids in green.

    3. Take the red trapezoids (that's the simplest case!):

    Point A must be situated on a parallel to the x-axis with the equation y = 8.

    The area of a trapezoid is calculated by : A_t = \frac{|\overline{AB}|+|\overline{CD}|}{2} \cdot h

    |\overline{CD}| = 8, h = 8

    So you have to solve for |\overline{AB}|:

    24 = \frac{|\overline{AB}|+8}{2} \cdot 5

    which yields |\overline{AB}|= 1.6

    So you get 2 different points A: A_1(-0.6, 8) or A_2(2.6, 8)

    4. The next 4 trapezoids are much more difficult to determine.

    From the book where I got this problem has answer but does not have a solution. It says A(-1,8). I wonder how they got it.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jun 2009
    Posts
    660
    Thanks
    133

    Re: What are the coordinates of A

    To begin with, I think that the answer given in your book is wrong.

    I think that the wording, 'a trapazoid ABCD', implies that the points should be taken in that order in which case the point A lies between the points B and D (and outside of the triangle BCD).
    With that assumption, there seem to be two possible constructions, one where AB is parallel to DC and the other where AD is parallel to CB. The first of these is the easier to deal with.

    The triangle BCD has a base of length 8 and a height 5 so its area is 8*5/2=20. That means that the triangle ABD should have an area of 4.
    Take AB to be the base of ABD, then its height will be 5 in which case its area will be 5*AB/2 and that has to equal 4.
    From that it's easy to calculate the x coordinate of A, (and it agrees with an answer given in an earlier post).

    The other possible position for A is more awkward to calculate, and since your book gives only one answer, I would guess that you are not expected to find it.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,829
    Thanks
    123

    Re: What are the coordinates of A

    Quote Originally Posted by rcs View Post
    From the book where I got this problem has answer but does not have a solution. It says A(-1,8). I wonder how they got it.
    Same to me!

    If A(-1, 8) then |\overline{AB}| = 2.

    With |\overline{CD}| = 8

    The area of the trapezoid is: A_T = \frac{2+8}2 \cdot 5=5 \cdot 5 = 25

    But maybe this result includes the VAT
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,707
    Thanks
    626

    Re: What are the coordinates of A

    Hello, rcs!

    Is there a typo in the problem?
    That would explain their answer.


    A trapezoid ABCD has an area of 25 sq. units.
    Three of its coordinates are: B(1,8), C(2,3), and D(-6,3) .
    Find the coordinates of A.

    The problem is not clearly stated.
    As earboth pointed out, there are six possible trapezoids.

    I assume that the horizontal side CD is the "base"
    . . and AB \parallel CD.
    Hence, A is in Quadrant 2.

    Code:
                        |
                  (x,8) |   (1,8)
                   A♥ * * * * ♥B
                  *     |      *
                *       |       *
              *         |        *
            *           |         *
         D♥ * * * * * * |* * * * * ♥C
       (-6,3)           |         (2,3)
                        |
      - - - - - - - - - + - - - - - - - -
                        |
    The height is h = 5.

    The bases are: . C\!D \,=\, 8,\;AB \,=\, 1-x

    Area of a trapezoid: . A \:=\:\tfrac{h}{2}(b_1 + b_2)

    Hence, we have: . \tfrac{5}{2}\big[8 + (1-x)\big] \:=\:2{\color{red}5}

    . . \tfrac{5}{2}(9 -x) \:=\:25 \quad\Rightarrow\quad 9-x \:=\:10 \quad\Rightarrow\quad x \:=\:-1

    Therefore: . A(\text{-}1,8)
    Follow Math Help Forum on Facebook and Google+

  7. #7
    rcs
    rcs is offline
    Senior Member rcs's Avatar
    Joined
    Jul 2010
    From
    iligan city. Philippines
    Posts
    455
    Thanks
    2

    Re: What are the coordinates of A

    Yeah I gues... Thanks sir Soroban
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: March 28th 2011, 10:40 PM
  2. Replies: 10
    Last Post: September 24th 2010, 04:33 AM
  3. Replies: 6
    Last Post: August 30th 2010, 03:34 PM
  4. Covert Cartesian coordinates eq. into Cylindrical coordinates
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: August 6th 2010, 06:56 AM
  5. Replies: 6
    Last Post: February 4th 2009, 12:12 AM

Search Tags


/mathhelpforum @mathhelpforum