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In $\triangle ABC$, shown here, $\overline{AB}$ and $\overline{AC}$ have each been divided into four congruent segments. What fraction of triangle $ABC$ is shaded? Express your answer as a common fraction.


https://latex.artofproblemsolving.com/7/3/4/734d6369be3ad3401903d79373fba68f790af690.png

 Oct 15, 2019
 #1
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Let the area  of  triangle ABC  =  A

 

Discounting the bottom unshaded area the remaining triangle  is  (3/4)^2 * area of ABC  =

 

(9/16) A       

 

So....the area  of the bottom unshaded area  = A  - (9/16)A  =  (7/16)A      (1)

 

Looking at  the "top" triangle  composed of   the top shaded area and the underneath unshaded area......the area of this triangle   =  (1/2)^2 * area of ABC  =  (1/4)A

 

And the area of the top shaded area  =  (1/4)^2 * area of ABC  =  (1/16)A

 

So....the   area  of the second unshaded region  =   (1/4)A - (1/16)A  =  (3/16)A    (2)

 

So.....the area of the shaded regions  =

 

area of ABC   - the areas of the two unshaded regions  = 

 

area of ABC  -  (1) - (2)  =

 

A   -   (7/16)A  -  (3/16)A  =

 

(6/16)A  =  (3/8) A

 

So....the shaded areas  are (3/8)  the area of  triangle ABC

 

 

cool cool cool

 Oct 15, 2019

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