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Inside the 7x8 rectangle below, one point is chosen a distance  sqrt(2) from the left side and a distance sqrt(7)  from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.

 

 

-reva *i need explanantion--- i am stuck after all the given info, how do you calculate area of shaded region. thnx bestiessssssssssss save the turtles

 Nov 12, 2019
 #1
avatar+2490 
+1

Im confused which side is 7 and which side is 8.

 

Can we assume that the left and right is 7 and the top and bottom is 8?

 Nov 12, 2019
 #2
avatar+94 
0

Calc user can you answer my question? https://web2.0calc.com/questions/question-i-need-the-answer-to

VooFIX  Nov 12, 2019
 #3
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0

yes the smaller side is 7 and and the bigger side is 8

 Nov 12, 2019
 #4
avatar+2490 
0

Does this make sense I tried solving it but couldn't (I was confused how x can be root 7 distance away from BC)

 Nov 12, 2019
edited by CalculatorUser  Nov 12, 2019
 #5
avatar
+2

 

 

Area of a Triangle equals one-half the base times the height ........                                     A  = 1/2 x b x h

 

Bottom triangle:  Base is the long side of the rectangle, height is the sqrt(7)                       AB  =  1/2 x 8 x sqrt(7)

 

Top triangle:  Base is the long side of the rectangle, height is [short side minus sqrt(7)]      AT  =  1/2 x 8 x [7 – sqrt(7)]

 

Calculate the areas of the two triangles and add them together. 

 

1/2 x 8 x sqrt(7)               =              + 4 sqrt(7)

1/2 x 8 x [7 – sqrt(7)]       =         28 – 4 sqrt(7) 

add together                   =         28 

 

Hmmm.  This makes me wonder.  If the lines inside the rectangle were made of rubber bands so that you could move their connection point anywhere around in there, would the area of the shaded part always be half of the area of the rectangle?  I'm not saying that would happen, I'm just wondering it because the area came out half in this instance.

 

 

 

 

 

.

 Nov 12, 2019
 #7
avatar+105370 
0

Your conjecture is correct

 

Let  the point be   x units from the top of the rectangle

 

Area  of top triangle  =  (1/2) (8) (x)

 

Area of bottom triangle = (1.2)(8) ( 7 - x)  =

 

Total of two areas =  (1/2)(8) ( x + 7 - x)  =  (1/2)(8) (7)  = 28 units^2

 

 

cool smileycool

CPhill  Nov 12, 2019
 #6
avatar+105370 
+2

The height of the shaded triangle at the top can be expressed as  (7 - √7) units

And  its base can be expressed as  8  units

 

The height of the shaded tiangle at the bottom can be expressed as √7 units

And its base can be expressed as 8

 

So.....the area of the two regions is just

 

(1/2) (8)  [ 7 - √7  + √7 ]  units  =

 

4 * 7   =

 

28 units^2

 

 

cool cool cool

 Nov 12, 2019
 #8
avatar+2490 
+1

oh ok it makes sense to me now I didn't know how the point could be place sqrt(7) from the bottom side, it confused me.

CalculatorUser  Nov 12, 2019

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