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Find the shaded area.

Note: The shaded area is a rectangle.

 

 Jan 10, 2020
 #1
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6 ,

The area of parallelogram is b*h=3*4=12 , Rectangle inside parallelogram and on its 2 sides (parralel sides) therefore equal to 1/2 its are so 1/2*12=6 

 Jan 10, 2020
 #2
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In the first place, the base of the rectangle is 6, not 3.  Second, I dispute the assumption that the inner rectangle is half the area of the larger. 

Guest Jan 10, 2020
 #3
avatar+21707 
+1

Guest is referrring to the parallellgram which contains the rectangle  ....not the lareger 6 x 4 rectangle

   so area of the parallelogramI IS base x height   = 3  x 4 = 12

 

    I do not know about the second part of their solution though it may be correct.....

ElectricPavlov  Jan 10, 2020
 #4
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Oh, I see what you mean now.  Thanks for that.  However, wouldn't the base of that parallelogram, as a hypotenuse, equal the square root of 32+42 ?  I'm just not getting it, maybe I ought to just move on.  Thanks again for the parallelogram explanation.

Guest Jan 10, 2020
 #5
avatar+21707 
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Yah....you could use that as the base (it would be 5) , but then you would have to calculate the height (which is NOT 3)...

     ___________________    ___

    /                                   /         |

  /                             3    /           h

/                                   /             |

________5_________        ___|__

ElectricPavlov  Jan 10, 2020
 #7
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OMG I am so dense.  You're calling the side of the parallelogram that's 3 units long the base.  It FINALLY makes sense to me.  Thanks for your patience and persistence.  And to think I was Mrs Cochran's favorite geometry student. 

Guest Jan 11, 2020
 #6
avatar+109061 
+1

The triangle on the left forms  a 3 - 4 - 5 right triangle

 

Note that the area of  this triangle  = (1/2) (3)(4)  =   6

 

And drawing a segment from the lower left vertex of the large rectangle perpendicular  to the side of the shaded rectangle  will also form an altitude of this right triangle

 

So

 

(1/2) (base) (altitude)  = 6

(1/2) (5) (altitude) = 6

(5/2) ( altitude) = 6

altitude = 12/5 = 2.4

 

And this perpendicular  also foms a right triangle  with a hypotenuse of  3  and one leg =  2.4

So....the other leg =  sqrt (3^2 - 2.4^2 ) = 1.8

 

And the area  of this triangle  =  (1/2) (product of the leg lengths) =

  (1/2) (2.4)(1.8) =    2.16 units^2

 

And, by AAS, this triangle is congruent to the   smaller right triangle at the  lower  right of the  figure

 

And we  have two of these smaller right  triangles and  two of the larger right triangles

 

So....their  total area =  2 ( 6 +  2.16) =  2 ( 8.16)  = 16. 32

 

And the area  of the large rectangle =   6 * 4 =  24

 

So.....the shaded area =  24 - 16.32  =  7.68 units^2

 

 

cool cool cool

 Jan 10, 2020
edited by CPhill  Jan 10, 2020

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