Results 1 to 4 of 4

Math Help - [SOLVED] trapezoid problem area

  1. #1
    Newbie
    Joined
    Jun 2008
    Posts
    2

    [SOLVED] trapezoid problem area

    this is a tough problem and i just can't seem to solve it. please help, thank you

    there is a trapezoid EFZD. top left is E, top right is F, bottom right is Z, and bottom left is D. it is an isosceles trapezoid. the intersection of the diagonals EZ and FD is at point Q. the trapezoid is the lateral face of a frustum and the frustum has base areas of 20 and 32. triangle EQF has an area of 12. find the area of the trapezoid EFZD. we don't know the shape of the bases of the frustum. the bases can be square, triangular, hexagonal, or anything else.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by roxmysox View Post
    this is a tough problem and i just can't seem to solve it. please help, thank you

    there is a trapezoid EFZD. top left is E, top right is F, bottom right is Z, and bottom left is D. it is an isosceles trapezoid. the intersection of the diagonals EZ and FD is at point Q. the trapezoid is the lateral face of a frustum and the frustum has base areas of 20 and 32. triangle EQF has an area of 12. find the area of the trapezoid EFZD. we don't know the shape of the bases of the frustum. the bases can be square, triangular, hexagonal, or anything else.
    I've sketched the trapezium.

    1. The base area B and the top area T are similar. Therefore the proportion of these areas is:

    \frac BT = \frac{20}{32} = \left( \frac{EF}{DZ}\right)^2

    2. The triangles EFQ and DZQ are similar and therefore their areas have the same proportion as the areas B and T. This proportion is valid for every pair of corresponding length in the two triangles. Therefore

    \left( \frac{x}{h-x}\right)^2 = \frac{20}{32}

    3. The equation to calculate the area of triangle EFQ is:

    \frac12 \cdot EF \cdot x = 12

    4. The area of the trapezium is:

    A_T=\frac12 \cdot (EF+DZ) \cdot h

    from #2 you know: h = \sqrt{\frac85} \cdot x + x = x \left(\sqrt{\frac85} + 1\right)

    from #1 you know: DZ = \sqrt{\frac85} \cdot EF

    5. Plugin these terms into the equation of #4:

    A_T=\frac12 \cdot \left(EF+\sqrt{\frac85} \cdot EF \right) \cdot x\left(\sqrt{\frac85} + 1\right)

    A_T=\frac12 \cdot EF \left(1+\sqrt{\frac85} \right) \cdot x\left(\sqrt{\frac85} + 1\right)

    A_T=\underbrace{\frac12 \cdot EF  \cdot x}_{= 12} \left(\sqrt{\frac85} + 1\right)^2 ....... according to #3

    A_T=12 \left(\sqrt{\frac85} + 1\right)^2~\approx~ 61.55
    Attached Thumbnails Attached Thumbnails [SOLVED] trapezoid problem area-trapez_abmessungen.gif  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2008
    Posts
    2
    thanks so much.
    but one thing, i don't understand how you get <br />
DZ = \sqrt{\frac85} \cdot EF<br />
and <br />
h = \sqrt{\frac85} \cdot x + x = x \left(\sqrt{\frac85} + 1\right)<br />
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by roxmysox View Post
    thanks so much.
    but one thing, i don't understand how you get <br />
DZ = \sqrt{\frac85} \cdot EF<br />
and <br />
h = \sqrt{\frac85} \cdot x + x = x \left(\sqrt{\frac85} + 1\right)<br />
    to #1:
    \frac BT = \frac{20}{32} = \left( \frac{EF}{DZ}\right)^2

    \left( \frac{EF}{DZ}\right)^2 = \frac{20}{32} = \frac58

    \left( \frac{DZ}{EF}\right)^2 = \frac85

     \frac{DZ}{EF} = \sqrt{\frac85}~\iff~ DZ = \sqrt{\frac85} \cdot EF

    to #2:
    \left( \frac{x}{h-x}\right)^2 = \frac{20}{32}=\frac58

    \left( \frac{h-x}{x}\right)^2 =\frac85

     \frac{h-x}{x} =\sqrt{\frac85}

    h-x= \sqrt{\frac85} \cdot x

    h= \sqrt{\frac85} \cdot x + x = x\left(\sqrt{\frac85} + 1 \right)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Area of Trapezoid
    Posted in the Geometry Forum
    Replies: 7
    Last Post: December 14th 2010, 04:49 AM
  2. Area of a Trapezoid
    Posted in the Geometry Forum
    Replies: 2
    Last Post: May 5th 2010, 04:52 AM
  3. area of trapezoid
    Posted in the Geometry Forum
    Replies: 2
    Last Post: November 22nd 2009, 04:26 AM
  4. [SOLVED] Please help! 2 Trapezoid problem
    Posted in the Geometry Forum
    Replies: 1
    Last Post: May 4th 2009, 07:51 AM
  5. Area of Isosceles Trapezoid
    Posted in the Geometry Forum
    Replies: 1
    Last Post: May 3rd 2009, 04:44 PM

Search Tags


/mathhelpforum @mathhelpforum