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avatar+2552 

Given trapezoid ABCD with the measure of \(\angle{D}\) (in degrees) equals 45, AD = \(8\sqrt{2}\), AB = 4BC = 10. Find the area of the trapezoid.

 

     (image is not to scale)

 

 

I used trigonometry, and I got the answer of 88 (please check). However, this problem shouldn't use trig and shouldn't use the calculator. Can someone tell me how to solve it?

 Jul 13, 2019
 #1
avatar+8852 
+5

Draw a line segment from  A  perpendicular to  DC  that meets  DC  at point  E , and

draw a line segment from  B  perpendicular to  DC  that meets  DC  at point  F , like this:

 


 

The sum of the interior angles in  △ADE  =  180°

45°  +  90°  +  m∠DAE   =   180°

m∠DAE   =   180° - 45° - 90°

m∠DAE   =   45°

 

△ADE  is an isosceles triangle and the sides opposite the base angles are congruent.

DE  =  AE

 

And if   DE = b   then   AE = b

 

By the Pythagorean Theorem,

DE2 + AE2  =  (8√2)2

b2 + b2  =  (8√2)2

2b2   =   128

b2   =   64

b   =   8

 

ABFE  is a rectangle so

BF  =  AE  =  8

EF  =  AB  =  4

 

By the Pythagorean Theorem,

BF2 + FC2  =  102

82 + FC2  =  102

FC2  =  102 - 82

FC2  =  36

FC  =  6

 

Now we know:

 

AB  =  4

So we can say  base1 =  4

 

DE  =  8   and   EF  =  4   and   FC  =  6

So we can say  base2  =  8 + 4 + 6  =  18

 

AE  =  8

So we can say  height  =  8

 

area of trapezoid  =  (1/2)(base1 + base2)(height)

area of trapezoid  =  (1/2)(4 + 18)(8)

area of trapezoid  =  88

 

If you have the 45-45-90 triangle memorized, you can immediately recognize that DE and AE  must be 8,

but if not you can always figure it out this way.  smiley

 Jul 13, 2019
 #2
avatar+2552 
+3

Thank you! This should've been easy, I will have to remember about isosceles triangles and Pythagorean theorem.

CalculatorUser  Jul 15, 2019

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