Fill in the blanks!

To fill in the blanks using the Pythagorean theorem, we'll start with the basic equation:

\[ a^{2} + b^{2} = c^{2} \]

In your case with the triangle:

• Side A is given as 2 inches (let's call this side \( a \)),
• Side B is given as 3 inches (let's call this side \( b \)),
• Side C is the hypotenuse (let's call it \( c \)).

We can start by filling in the blanks with what we know:

1. \( a = 2 \) and \( b = 3 \): \[ 2^{2} + 3^{2} = c^{2} \] So we have: \[ 4 + 9 = c^{2} \] This means: \[ 13 = c^{2} \]

2. Filling in the blanks:

   • From \( 3^{2} + _^{2} = c^{2} \), we know \( b = 4 \): \[ 3^{2} + 4^{2} = c^{2} \]
   • The blank for c will now be: \[ _ + 4 = c^{2} \quad \text{(unknown)} \Rightarrow \text{with our earlier calculations, we fill in } 9. \]
   • Therefore: \[ _ = c^{2} \quad \text{is } 13. \]
   • Finally, substituting for hypotenuse c: \[ (underlined) + \sqrt{13} = c \quad \text{(the underlined part is 0)} \] Saying: \[ 0 + \sqrt{13} = c \Rightarrow c = \sqrt{13}. \]

Putting everything together:

• Underlined part = 9 (from \( 3^{2} + 4^{2} = c^{2} \)),
• First blank = 4 (since \( b = 4; 3^{2} + b^{2} = c^{2} \)),
• Last blank in the triangle = 13

So, the filled in parts are:

1. \( 4 \)
2. \( 9 \)
3. \( 13 \)

Thus it gives us: Triangle: A = 2 in. B = 3 in. C = \sqrt{13}