What is the area of the dark square? Help please

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The side of the square is 10cm and M,N,P and Q(wrongly named "D" under the real point D) are medium points. Find the dark square

PS: I know the square have 100cm² and I have found the two triangles DCN and DAB as 25cm² each, so we have 2 trapezius remaining to find the dark square area, that's it. thanks!

1 Answer
CW
Jan 12, 2018

shaded area #=20 " cm"^2#

Explanation:

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given #NC=5, CD=10, => DN=sqrt(5^2+10^2)=sqrt125#
Let #RS=x#
draw a line #ST#, parallel to #CD#, as shown in the figure.
#DeltaRST and DeltaCDN# are similar,
#=> (RS)/(ST)=(CD)/(DN)#,
#=> x/5=10/sqrt125#,
#=> x=(5xx10)/sqrt125=50/sqrt125#
shaded area #=x^2=50^2/125=20 " cm"^2#

Sol.2 )
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#DeltaRST, DeltaRDP,DeltaSCN, DeltaVBM and DeltaUAQ# are congruent,
let area of #DeltaRST=a#,
#=> "total area of the 5 congruent traingles" = 5a#
Trapeziuses #TUVS, QURD, PRSC, NSVB and MVUA# are congruent,
let area of trapezius #TUVS# be #b#,
#=> "total area of the 5 congruent trapeziuses" = 5b#
Area of squra #ABCD=10xx10=100#,
#=> 5(a+b)=100#,
#=> a+b=20#
#=> "shaded area" = a+b=20 " cm"^2#

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