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# 2015 NS 25

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3 Answer: 256/17

Mar 16, 2019

#1
+1

I tried setting the side length to a. Then, I used the length given and got an equation for the third small triangle. Then I got confused on how to solve that. Is there any other way?

Mar 16, 2019
#2
0

Call the side of the square, S

So...the area of the square = S^2

The area of the small triangle at the bottom left  = S* [ sqrt (4^2 - S^2) ] /2 / 2      (1)

The area of the small triangle at the top right = S* [ sqrt (5^2 - S^2)] /2        (2)

And the area of the small triangle at the top left =   [ S  - sqrt (4^2 - S^2)] [ S - sqrt (5^ - S^2) ] / 2     (3)

And the area of the right triangle = 6     (4)

So

(1) + (2) + (3) + (4)  =  S^2

S* [ sqrt (16 - S^2 ] /2  + S* [ sqrt (25 - S^2)] / 2   + [ S  - sqrt (16 - S^2)] [ S - sqrt (25 - S^2) ] / 2 + 6 = S^2

S* [ sqrt (16 - S^2 ]   + S* [ sqrt (25 - S^2)] + [ S  - sqrt (16 - S^2)] [ S - sqrt (25 - S^2) ]  + 12 = 2 S^2

S* [ sqrt (16 - S^2 ]   + S* [ sqrt (25 - S^2)] + [ S  - sqrt (16 - S^2)] [ S - sqrt (25 - S^2) ]  + 12 = 2 S^2

S* [ sqrt ( 16 - S^2) ] + S*[ sqrt (25 - S^2) ]  + S^2  - S*[ sqrt (16 - S^2)] - S* [ sqrt (25 - S^2)] +

sqrt [ ( 16 - S^2)  (25 - S^2) ] + 12  = 2S^2

sqrt [ (16 - S^2) (25 - S^2)] =  S^2 - 12       square both sides

(16 - S^2) ( 25 - S^2)  =  S^4 - 24S^2 + 144

400 - 41S^2 + S^4  = S^4 - 24S^2 + 144

17S^2  - 256  = 0

17S^2 = 256

S^2  =   256 / 17   Mar 16, 2019
#3
0

Thanks!!!

dgfgrafgdfge111  Mar 18, 2019