ANSWER :
1. 2ab + c^2
2. (a + b)^2 or (a^2 + 2ab + b^2)
3. 2ab
EXPLANATION :
Recall that the area of a triangle is 1/2 base x height.
and the area of a square is the square of its side.
From the problem, we have 4 congruent triangles.
The area of four triangles is :
[tex]\begin{gathered} A=4\times\frac{1}{2}(ab) \\ A=2ab \end{gathered}[/tex]
The area of the square is :
[tex]A=c^2[/tex]
Then the total area will be :
[tex]2ab+c^2[/tex]
It is equivalent to the area of the whole square in which the side is (a + b)
So that's :
[tex]A=(a+b)^2\quad or\quad(a^2+2ab+b^2)[/tex]
Since both expressions are equal, we can equate them :
[tex]\begin{gathered} 2ab+c^2=a^2+2ab+b^2 \\ \text{ Subtracting 2ab from both sides :} \\ 2ab+c^2-2ab=a^2+2ab+b^2-2ab \\ c^2=a^2+b^2 \end{gathered}[/tex]
