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# area

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Line segments are drawn from the vertices of the large square to the midpoints of the opposite sides to form a smaller, white square.

If each red or blue line-segment measures 10 m long, what is the area of the smaller, white square in m^2?

May 3, 2020

#1
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Note that  at the top of the figure we have a right triangle with two legs  of  10 and 20

So the area of this  right triangle  = (1/2) (10)(20)   = 100

And the base of this triangle  = sqrt  [ 10^2 + 20^2]  =  sqrt [ 500] =  10sqrt(5)

So......we can find the height of this triange as

100  = (1/2) (10sqrt (5) ) height

20 / sqrt (5)  =  height  =  4sqrt (5)

And using similar triangles  the height  of the small right triangle  on the top left  =  1/2  of this  = 2sqrt (5) = sqrt (20)

And the base  of this triangle  =  sqrt [ 10^2 - (sqrt(20))^2 ]  =  sqrt [ 100 - 20]  = sqrt (80)

So  the area of this small right triangle  =  (1/2)sqrt (20)sqrt(80) = (1/2)sqrt(1600) = (1/2) 40  = 20

And we have two of these triangles in the larger right triangle at the top

So....the area of the remaining   trapezoid  = 100 - 2(20)  = 60

So....we have   4 smaller right triangles and  4 trapezoids....so their combined areas  =

4 ( 20 + 60)  = 4 (80)  = 320      (1)

So....the area  of the white  square  =

area of the square  -  (1)   =

20^2  - 320  =

400 - 320  =

80   (m^2)

May 3, 2020