Step-by-step explanation:
Pythagoras' theorem for the smallest one :
[tex]c^2 = 4^2 + 6^2[/tex]
[tex]c^2 = 16 + 36[/tex]
[tex]c^{2}[/tex] = 52
Pythagoras' theorem for the middle one :
[tex]b^{2}[/tex] = [tex]6^{2}[/tex] + [tex]a^{2}[/tex]
Pythagoras' theorem for the biggest one :
[tex](4+a)^2 = c^2 + b^2[/tex]
[tex]16 + 8a + a^2 = 52 + b^2[/tex]
Using the formula before (for [tex]b^2[/tex]) it becomes :
[tex]16 + 8a + a^2 = 52 + (6^2 + a^2)[/tex]
[tex]16 + 8a + a^2 = 52 + 6^2 + a^2[/tex]
[tex]16 + 8a = 52 + 6^2[/tex]
16 + 8a = 52 + 36
16+8a = 88
8a = 88-16
8a = 72
[tex]a = \frac{72}{8}[/tex]
a = 9
Verifying :
[tex]b^2 = 6^2 + a^2[/tex]
[tex]b^2 = 36 + 81[/tex]
[tex]b^2 = 117[/tex]
[tex]b^{2}[/tex] = 117
The biggest one :
[tex](4+a)^2 = c^2 + b^2[/tex]
[tex](4+9)^2 = 52 + 117[/tex]
[tex]13^2 = 169[/tex]
True
