Question 645432
abcd is a trapezium in which ab parallel cd.bc and ad are the non parallel sides.it is given that ab=75cm,bc=42cm,cd=30cm and ad=39cm.Find the area of the trapezium

<pre>
{{{drawing(400,3600/17,-5, 80, -5,40,locate(37.5,0,75), locate(34,37,30),
locate(0,0,A), locate(75,0,B), locate(50,37,C), locate(19.8,37,D), 
line(0,0,75,0), line(0,0,19.8,33.6), locate(63,20,42),locate(6,20,39),line(19.8,33.6,49.8,33.6),line(49.8,33.6,75,0)   )}}}

Draw DE and CF perpendicular to AB.

Let AE = x and let BF = y, and DE = CF = h, and EF = CD = 30

{{{drawing(400,3600/17,-5, 80, -5,40,locate(34,0,30), locate(34,37,30),
locate(0,0,A), locate(75,0,B), locate(50,37,C), locate(19.8,37,D), 
line(0,0,75,0), line(0,0,19.8,33.6), locate(19.8,0,E), locate(47.5,18,h), locate(49.8,0,F),locate(63,20,42),locate(6,20,39),line(19.8,33.6,49.8,33.6),line(49.8,33.6,75,0), locate(9.9,0,x),locate(60,0,y), locate(20.5,18,h),
green(line(19.8,0,19.8,33.6),line(49.8,0,49.8,33.6))

   )}}}

Since AB = 75, x + 30 + y = 75
                    x + y = 45


By the Pythagorean theorem

 x² + h² = 39²
 y² + h² = 42²

 x² + h² = 1521
 y² + h² = 1764

Multiply the first equation by -1 and add the two equations:

-x² - h² = -1521
 y² + h² =  1764
----------------
 y² - x² = 243

Factorise:

(y - x)(y + x) = 243, and since x + y = 45, y + x = 45
   (y - x)(45) = 243
     45(y - x) = 243
        y - x = {{{243/45}}}
        y - x = 5.4

Add these two equations:

 y + x = 45
 y - x =  5.4
-------------
2y     = 50.4
     y = 25.2

Substitute in:

     y² + h² = 1764
(25.2)² + h² = 1764
 635.04 + h² = 1764
          h² = 1128.96
           h = 33.6

Formula for area of trapezium

A = (base<sub>1</sub> + base<sub>2</sub>)(height)/2

A = (75 + 30)(33.6)/2

A = 1764 cm²  

Edwin</pre>