Results 1 to 10 of 10
Like Tree5Thanks
  • 1 Post By skeeter
  • 1 Post By Idea
  • 1 Post By johng
  • 2 Post By Idea

Thread: inscribed quadrilateral

  1. #1
    Junior Member
    Joined
    Aug 2016
    From
    U.K.
    Posts
    33

    inscribed quadrilateral

    hi i have this:

    circle (x+4)^2 + (y+4)^2 = 20crosses the x-axis at the points A and B and y-axis at points C and D. Find the area of the quadrilateral ABCD.

    I spend so much time trying to figure height etc. but all i could get was sides lenghts, so the only way i could get to the solution was by using this quadrilateral formula
    x1 * y2 - y1 * x2 .... but as am doing this from textbook that way is never mentioned anywhere , so can someone please tell what would be the other way of doing it ?
    Last edited by AbYz; Feb 26th 2017 at 08:01 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,041
    Thanks
    3621

    Re: inscribed quadrilateral

    $(x+4)^2 + (0+4)^2 = 20 \implies x=-6 \text{ and } x=-2$

    $x=-6 \implies 4 + (y+4)^2 = 20 \implies y=0 \text{ and } y=-8$

    rectangle base = $|-2 - (-6)| = 4$

    rectangle height = $|0 - (-8)| = 8$
    Attached Thumbnails Attached Thumbnails inscribed quadrilateral-rect_circ.png  
    Thanks from AbYz
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    696
    Thanks
    316

    Re: inscribed quadrilateral

    inscribed quadrilateral-circlequad.jpg
    Thanks from AbYz
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    1,108
    Thanks
    460

    Re: inscribed quadrilateral

    Sometimes quadrilateral means that the polygon is convex. I prefer to say a quadrilateral is any polygon with 4 vertices, so edges can intersect at points other than the vertices. So there can be two quadrilaterals ABCD described by your problem, depending on how the vertices are labeled. Idea has given you one of these. The other is drawn below. Sometimes, it is helpful to know the answers. The convex quadrilateral has area 16 and the other quadrilateral has area 6. For either of these, it is helpful to have the formula for the distance from a point $(x_0,y_0)$ to a line with equation $Ax+By+C=0$, namely
    $${Ax_0+By_0+C\over \sqrt{A^2+B^2}}$$
    This helps to find heights.

    Edit: By the way, the formula for area that you mentioned is valid only for convex quadrilaterals.

    Last edited by johng; Feb 26th 2017 at 02:43 PM.
    Thanks from AbYz
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,041
    Thanks
    3621

    Re: inscribed quadrilateral

    well ... looks like I had rectangle on the brain. nevermind.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Aug 2016
    From
    U.K.
    Posts
    33

    Re: inscribed quadrilateral

    Quote Originally Posted by johng View Post
    Sometimes quadrilateral means that the polygon is convex...
    Basically im not on this level yet.. its the trapezoid in my case. But trying everything i got a lot similiar values to 16 , they just differ by decimal places. only the formula i mentioned before gave me exact 16 , not sure why.
    On the other note, could i just calculate bases and then calculate the 2 triangles on sides and by doing that i autamatically get trapezs height right ?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    1,108
    Thanks
    460

    Re: inscribed quadrilateral

    A convex quadrilateral is just a quadrilateral Q such that for any 2 points "inside" Q, the line segment joining the 2 points lies entirely inside Q. The first quadrilateral is convex.



    Now this quad is not convex; choose 2 points inside of the 2 triangles. The line connecting the 2 points does not stay inside the quad. You do the computations as indicated. If you have trouble, post again.

    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,198
    Thanks
    2626
    Awards
    1

    Re: inscribed quadrilateral

    Quote Originally Posted by johng View Post
    Now this quad is not convex; choose 2 points inside of the 2 triangles. The line connecting the 2 points does not stay inside the quad. You do the computations as indicated. If you have trouble, post again.
    @johng, I don't know anyone actively involved in axiomatic geometry who thinks of your second figure as a quadrilateral. In fact going back to Hilbert, E H Moore, R L Moore, the four segments were required to be non-intersecting. It is true that some modern authors have defined a figure called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral.

    Moreover, a careful reading of the original question gives $A,~B,~C,~\&~D$ in that particular order. Those in the editing community would expect the test taker to use $A: (-6,0),~B: (-2,0),~C: (0,-2),~\&~D: (0,-6)$ to the quadrilateral $ABCD$.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    696
    Thanks
    316

    Re: inscribed quadrilateral

    O(0,0)

    convex quad area = area OAC - area OBD = 18 - 2 = 16
    Thanks from johng and AbYz
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    1,108
    Thanks
    460

    Re: inscribed quadrilateral

    Idea, as usual you gave a simple clear solution. It makes my complicated solution look kind of stupid.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inscribed Quadrilateral
    Posted in the Geometry Forum
    Replies: 3
    Last Post: Oct 22nd 2011, 08:37 AM
  2. Quadrilateral inscribed in a circle
    Posted in the Geometry Forum
    Replies: 2
    Last Post: May 6th 2011, 04:42 AM
  3. Area of Circle with inscribed quadrilateral
    Posted in the Geometry Forum
    Replies: 3
    Last Post: Apr 4th 2011, 03:50 PM
  4. Quadrilateral inscribed in a Circle
    Posted in the Geometry Forum
    Replies: 22
    Last Post: Feb 17th 2010, 01:34 PM
  5. Quadrilateral Inscribed in a Circle?
    Posted in the Geometry Forum
    Replies: 1
    Last Post: Jun 10th 2009, 12:46 PM

/mathhelpforum @mathhelpforum