i think:
  
 the hyp of the bigger triangle is devided into two parts  lets say one of them is "m" and the other is 'n'
 so we can find m using pytha. theorem:
 \(10^2-6^2=8^2\\ m=8\)
  
 and there is a formula that:
 \(a^2=m*(m+n)\\ a=10\\ m=8\\ 10^2=8*(8+n)\\ n=\frac92 \)
  
 \(c=m+n\\ c=8+4.5=12.5\)
  
 let s use again pyth. theorem:
 \(12.5^2-10^2=7.5^2\\ b=7.5\)
  
 \(A_{triangle}=\frac{7.5*10}{2}=37.5\)
  
 another easier way to do that:
 \(h^2=m*n\\ 6^2=8*n\\ n=\frac92\)
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