Question 1166670
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            It was not simple task for me to find the way --- but I found it (!)


            Unfortunately,  the solution goes through  UGLY  calculations  --- but I do not see other way
            and therefore,  I was  FORCED  to go through it.



I will show the solution step by step.  
Watch my steps attentively.



<pre>
1)  We have right angled triangle ABE.  The hypotenuse AB is 61 units long; the leg AE is 60 units long.

    Hence, the leg BE is  {{{sqrt(61^2-60^2)}}} = 11.



2)  We have right angled triangle ADE.  The hypotenuse AD is 156 units long; the leg AE is 60 units long.

    Hence, the leg ED is  {{{sqrt(156^2-60^2)}}} = 144.



3)  The area of the triangle ABD is  {{{S[ABD]}}} = {{{(1/2)*BD*AE}}} = {{{(1/2)*(11+144)*60}}} = 4650 square units.



4)  Triangles ADE and BCD are similar (they are right-angled and have congruent angles DBC and BDA).

    From similarity, we have this proportion 

        {{{abs(DC)/abs(BD)}}} = {{{abs(AE)/abs(DE)}}},  or  {{{abs(DC)/(11+144)}}} = {{{60/156}}} = {{{5/13}}}.

        hence  |DC| = {{{(155*5)/13}}}.



5)  Then the area of the triangle BCD  is  {{{S[BCD]}}} = {{{(1/2)*DC*BD}}} = {{{(1/2)*(155*5/13)*155}}} = {{{120125/26}}}.




6)  The area of the trapezoid ABCD is the sum of areas of triangles  

        {{{S[ABCD]}}} = {{{S[ABD]}}} + {{{S[BCD]}}} = 4650 + {{{120125/26}}} = {{{(4650*26+120125)/26}}} = {{{241025/26}}}.




7)  From the similarity of triangles ADE and BCD we have this proportion

        {{{abs(AD)/abs(AE)}}} = {{{abs(BC)/abs(DC)}}},  or   {{{156/60}}} = {{{abs(BC)/((155*5)/13))}}}

    
    From the proportion,  |BC| = {{{(156*155*5)/(13*60)}}} = {{{(12*155*5)/60}}} = 155.




8)  Now the area of the trapezoid ABCD is half sum of its bases AD and BC multiplied by the height of the trapezoid, or

        {{{((156+155)/2)*H}}} = {{{241025/26}}}.


    It gives  H = {{{(2*241025)/(311*26)}}} = 59 {{{4976/(311*26)}}} = 59 {{{16/26}}} = 59 {{{8/13}}}.    <U>ANSWER</U>
</pre>

SOLVED.



I hope I deserved your &nbsp;"THANKS" &nbsp;for my efforts --- so I am open to accept them (!)