Squares

Cut out the square with the integer side from the square with the integer side so that the residual area is 100. What is the longest possible side of the larger square?

Correct answer:

a =  26

Step-by-step explanation:

$$\begin{gathered} a^2-b^2=100 \\ a>b>0 \\ {a}^2-b^2=100 \\ (a-b)\cdot (a+b)=100=2\cdot 2\cdot 5\cdot 5=2\cdot (2\cdot 5\cdot 5)=2\cdot 50 \\ a-b=2 \\ a+b=50 \\ a=5\cdot 5+1=26 \\ b=a-2=26-2=24 \\ \text{ Verifying Solution: } \\ {S}_1=a^2=26^2=676 \\ {S}_2=b^2=24^2=576 \\ d=S_1-S_2=676-576=100 \end{gathered}$$

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