Calculate a Side Measurement Using Pythagorean Theorem (A) - Free Printable

The provided image is a worksheet that uses the Pythagorean Theorem to calculate missing side lengths in right triangles. The Pythagorean Theorem states:

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

where:
- \(a\) and \(b\) are the lengths of the two legs of the right triangle,
- \(c\) is the length of the hypotenuse (the side opposite the right angle).

Explanation of Each Problem

#### Problem 1:
Given:
- One leg (\(a\)) = 6 mm
- Hypotenuse (\(c\)) = 8.49 mm
- Find the other leg (\(b\)).

Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
6^2 + b^2 = 8.49^2
\]
Calculate the squares:
\[
36 + b^2 = 72.0801
\]
Solve for \(b^2\):
\[
b^2 = 72.0801 - 36
\]
\[
b^2 = 36.0801
\]
Take the square root of both sides:
\[
b = \sqrt{36.0801}
\]
\[
b \approx 6 \, \text{mm}
\]

#### Problem 2:
Given:
- One leg (\(a\)) = 4.6 m
- Hypotenuse (\(c\)) = 6.1 m
- Find the other leg (\(b\)).

Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
4.6^2 + b^2 = 6.1^2
\]
Calculate the squares:
\[
21.16 + b^2 = 37.21
\]
Solve for \(b^2\):
\[
b^2 = 37.21 - 21.16
\]
\[
b^2 = 16.05
\]
Take the square root of both sides:
\[
b = \sqrt{16.05}
\]
\[
b \approx 4 \, \text{m}
\]

#### Problem 3:
Given:
- One leg (\(a\)) = 2.5 km
- Other leg (\(b\)) = 3.0 km
- Find the hypotenuse (\(c\)).

Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
2.5^2 + 3.0^2 = c^2
\]
Calculate the squares:
\[
6.25 + 9.00 = c^2
\]
\[
c^2 = 15.25
\]
Take the square root of both sides:
\[
c = \sqrt{15.25}
\]
\[
c \approx 4.55 \, \text{km}
\]

#### Problem 4:
Given:
- One leg (\(a\)) = 7.6 AU
- Hypotenuse (\(c\)) = 9.1 AU
- Find the other leg (\(b\)).

Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
7.6^2 + b^2 = 9.1^2
\]
Calculate the squares:
\[
57.76 + b^2 = 82.81
\]
Solve for \(b^2\):
\[
b^2 = 82.81 - 57.76
\]
\[
b^2 = 25.05
\]
Take the square root of both sides:
\[
b = \sqrt{25.05}
\]
\[
b \approx 5.00 \, \text{AU}
\]

#### Problem 5:
Given:
- One leg (\(a\)) = 9.6 mi
- Hypotenuse (\(c\)) = 14.01 mi
- Find the other leg (\(b\)).

Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
9.6^2 + b^2 = 14.01^2
\]
Calculate the squares:
\[
92.16 + b^2 = 196.2801
\]
Solve for \(b^2\):
\[
b^2 = 196.2801 - 92.16
\]
\[
b^2 = 104.1201
\]
Take the square root of both sides:
\[
b = \sqrt{104.1201}
\]
\[
b \approx 10.2 \, \text{mi}
\]

#### Problem 6:
Given:
- One leg (\(a\)) = 5.4 m
- Other leg (\(b\)) = 7 m
- Find the hypotenuse (\(c\)).

Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the known values:
\[
5.4^2 + 7^2 = c^2
\]
Calculate the squares:
\[
29.16 + 49 = c^2
\]
\[
c^2 = 78.16
\]
Take the square root of both sides:
\[
c = \sqrt{78.16}
\]
\[
c \approx 8.84 \, \text{m}
\]

Final Answers:

1. \( b \approx 6 \, \text{mm} \)
2. \( b \approx 4 \, \text{m} \)
3. \( c \approx 4.55 \, \text{km} \)
4. \( b \approx 5.00 \, \text{AU} \)
5. \( b \approx 10.2 \, \text{mi} \)
6. \( c \approx 8.84 \, \text{m} \)

\boxed{6 \, \text{mm}, 4 \, \text{m}, 4.55 \, \text{km}, 5.00 \, \text{AU}, 10.2 \, \text{mi}, 8.84 \, \text{m}}