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Rectangle ABCD is cut by two lines as shown. Find the area of AEXH.

 

https://latex.artofproblemsolving.com/6/6/c/66cb67543703cafc48a84fd49998af9ddbc7d9c1.png

 Dec 18, 2019
 #1
avatar+111321 
+1

Not the easiest way to solve this, I suspect

 

Note that  the distance from X  to AB = (1/2)AD  = (1/2) (9)  =  4.5 units

 

The line  containing  HF  has a slope  of  5/8

 

Extend BA  in the direction of A  and   FH  in the direction of  H  to  meet at  I

 

So..... the length of IA  can be found as

 

5/8  = 2/ IA

5/16 =  1/ IA

IA  =  16/5

 

So   the area  of triangle  IAH =  (1/2)(Base)(Height)  = (1/2)(IA)(HA)  =  (1/2)(16/5) (2)  =  16/5 =  3.2 units

 

And the area of triangle  XEI  =   (1/2) (Base)(Height)  = (1/2)(5 + 16/5) (4.5)  = 2.25 ( 41/5) = 18.45 units

 

So  [AEXH]   =  [ XEI ]  - [ IAH]   =  [ 18.45 ] - [3.2  ] =  15.25 units^2

 

Here's a pic :

 

 

 

cool cool cool

 Dec 18, 2019

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