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{Geometry} Find the Area of the quadrilateral

  1. Jul 22, 2015 #1
    1. The problem statement, all variables and given/known data
    http://i.imgur.com/lzTN7If.png
    Excuse the bad drawing

    Point E lies on the side AC of the square ACBD. The segment EB is broken up into 4 equal parts as well as the segment ED. If JK = 3 find the area of the quadrilateral FGKJ


    2. Relevant equations
    Trapezoid equation to find the height and the area. As well as pythagoras equation

    3. The attempt at a solution


    My attempt was I know that quadrilateral is a isosceles trapezoid because you have opposite parallel sides and two sides equal. I assumed since the segment JK was 3 FG was 1 and the side BD was 4. I also assumed E split AC in half. I tried to calculate the entire segment EB using 4^2+2^2=sqrt(20) I divided that by 4 to find length of one of the 4 equal segments. Then multiplied that by 2 to get sqrt(5). Used the formula for finding H of an isosceles trapezoid and got 2. Area then would be 1/2(4)(2)=4

    My geometry is weak so don't have confidence in the approach. Thanks for any help
     
    Last edited: Jul 22, 2015
  2. jcsd
  3. Jul 22, 2015 #2
    I do not know if the diagram you provide is your interpretation of a problem or the diagram given in a problem. However, as drawn, ACBD would not be a square, since the name implies that its sides are AC, CB, BD, and DA. Also, do you mean that you need the area of FGKJ?
     
  4. Jul 22, 2015 #3
    Yes picture is not drawn to reality this is error in my free hand ability. Yes I'm sorry that was a typo you are right FGKJ I'll fix that
     
  5. Jul 22, 2015 #4

    SteamKing

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    You really don't need much geometry to solve this problem except similar triangles.

    Applying the Pythagorean Theorem is nice, but unnecessary.
     
  6. Jul 22, 2015 #5
    The sides not being equal are not what I meant. When labeling a shape, the order in which you label describes the sides that are present. Usually, you want to label the shape clockwise or counterclockwise. So for ex. ABCD has sides AB, BC, CD, and DA. I was pointing out that your image, as drawn, shows a square ACDB, not ACBD. Your method seems to be correct, as the area is 4 as far as I can see. If it is an simpler for you, the question could be solved entirely with the pythagorean theorem, just splitting the trapezoid into 2 equal triangles and a rectangle. Also, can you justify your assumptions of E being the midpoint of AC, and that FG has length 1?
     
  7. Jul 22, 2015 #6
    Ah thank you sorry for the confusion.

    Your hint helped me also with similar triangles

    EJK is similar to EBD

    SO EJ/EB=EK/ED Therefore BD=EB/EJ *JK =4 Therefore area of EBD=8 then the area of EJK would be (3/4)^2 *8 and EFG would then be (1/4)^2 * 8 So then you take the difference of those to get the area of the quadrilateral which is 4.
     
  8. Jul 22, 2015 #7
    Exactly, similar triangles can be used to find the areas, which I believe was a hint from SteamKing and not myself. Also, as far as I can see, E does not need to be the midpoint of AC.
     
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