Question

Fill in the blanks!

Use the Pythagorean theorem to find the length of the hypotenuse, c.
a^(2) + b^(2) = c^(2)
3^(2) + _^(2) = c^(2)
_ + 4 = c^(2)
_ = c^(2)
(underlined)+ \sqrt(13) = c
\sqrt(13) = c

Traingle:
A
2in.
C
3in.
B
c = \sqrt(_)

Answer 1

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}