In order to calculate the value of each missing variable, we need to know that the area of a square is given by the length of its side squared.
So, for the yellow triangle, we have:
[tex]\begin{gathered} \text{Area}=\text{side}^2 \\ 9=\text{side}^2 \\ \text{side}=3 \end{gathered}[/tex]
For the blue square:
[tex]\begin{gathered} 16=s^2 \\ s=4 \end{gathered}[/tex]
Now, we can use the Pythagorean Theorem to find the value of b:
[tex]\begin{gathered} b^2=3^2+4^2 \\ b^2=9+16 \\ b^2=25 \\ b=5 \end{gathered}[/tex]
Now, for the area of the red square:
[tex]\begin{gathered} a=b^2 \\ a=5^2 \\ a=25 \end{gathered}[/tex]
Doing the same for the other figure, we have:
[tex]\begin{gathered} \text{pink:} \\ 144=s^2\to s=12 \\ \text{green:} \\ 81=s^2\to s=9 \end{gathered}[/tex]
The area of a right triangle is half the product of the legs, so let's calculate the missing leg (that is, the value of c) using the Pythagorean Theorem:
[tex]\begin{gathered} 12^2=9^2+c^2 \\ c^2=144-81 \\ c^2=63 \\ c=7.94 \\ \\ d=\frac{9\cdot7.94}{2} \\ d=35.73 \end{gathered}[/tex]
