Pythagorean Theorem Worksheets - Free Printable

The worksheet provided involves solving word problems using the Pythagorean Theorem, which states:

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

where \( a \) and \( b \) are the legs of a right triangle, and \( c \) is the hypotenuse (the longest side).

Let's solve each problem step by step.

---

Problem 1: Diagonal Length of the Television

Question: Gavin bought a new television for his apartment. It is 50 inches wide and 30 inches tall. Find the diagonal length of the television.

#### Solution:
1. Identify the sides of the right triangle:
- Width (\( a \)) = 50 inches
- Height (\( b \)) = 30 inches
- Diagonal (\( c \)) is what we need to find.

2. Apply the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
\[
50^2 + 30^2 = c^2
\]

3. Calculate the squares:
\[
50^2 = 2500
\]
\[
30^2 = 900
\]

4. Add the squares:
\[
2500 + 900 = 3400
\]

5. Solve for \( c \):
\[
c^2 = 3400
\]
\[
c = \sqrt{3400}
\]
\[
c \approx 58.31 \text{ inches}
\]

#### Final Answer:
\[
\boxed{58.31 \text{ inches}}
\]

---

Problem 2: Shortest Distance Heather Can Cover

Question: Heather is on her way home from school. She walks 3 miles North and then 4 miles East. What is the shortest distance she can cover to reach home the fastest?

#### Solution:
1. Identify the sides of the right triangle:
- Distance North (\( a \)) = 3 miles
- Distance East (\( b \)) = 4 miles
- Shortest distance (\( c \)) is what we need to find.

2. Apply the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
\[
3^2 + 4^2 = c^2
\]

3. Calculate the squares:
\[
3^2 = 9
\]
\[
4^2 = 16
\]

4. Add the squares:
\[
9 + 16 = 25
\]

5. Solve for \( c \):
\[
c^2 = 25
\]
\[
c = \sqrt{25}
\]
\[
c = 5 \text{ miles}
\]

#### Final Answer:
\[
\boxed{5 \text{ miles}}
\]

---

Problem 3: Length of the Other Leg of the Triangle

Question: Mr. Cooper tells you that a right triangle has a hypotenuse of 200 mm and a leg of 172 mm. Find the length of the other leg.

#### Solution:
1. Identify the sides of the right triangle:
- Hypotenuse (\( c \)) = 200 mm
- One leg (\( a \)) = 172 mm
- The other leg (\( b \)) is what we need to find.

2. Apply the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
\[
172^2 + b^2 = 200^2
\]

3. Calculate the squares:
\[
172^2 = 29584
\]
\[
200^2 = 40000
\]

4. Substitute and solve for \( b^2 \):
\[
29584 + b^2 = 40000
\]
\[
b^2 = 40000 - 29584
\]
\[
b^2 = 10416
\]

5. Solve for \( b \):
\[
b = \sqrt{10416}
\]
\[
b \approx 102.06 \text{ mm}
\]

#### Final Answer:
\[
\boxed{102.06 \text{ mm}}
\]

---

Problem 4: Distance from the Base of the House for the Ladder

Question: How far apart from the base of the house should an 18-foot ladder be placed so that it exactly reaches the top of a 15-foot wall?

#### Solution:
1. Identify the sides of the right triangle:
- Hypotenuse (\( c \)) = 18 feet (length of the ladder)
- One leg (\( a \)) = 15 feet (height of the wall)
- The other leg (\( b \)) is the distance from the base of the house to the foot of the ladder, which we need to find.

2. Apply the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
\[
15^2 + b^2 = 18^2
\]

3. Calculate the squares:
\[
15^2 = 225
\]
\[
18^2 = 324
\]

4. Substitute and solve for \( b^2 \):
\[
225 + b^2 = 324
\]
\[
b^2 = 324 - 225
\]
\[
b^2 = 99
\]

5. Solve for \( b \):
\[
b = \sqrt{99}
\]
\[
b \approx 9.95 \text{ feet}
\]

#### Final Answer:
\[
\boxed{9.95 \text{ feet}}
\]

---

Summary of Answers:

1. Diagonal length of the television: \(\boxed{58.31 \text{ inches}}\)
2. Shortest distance Heather can cover: \(\boxed{5 \text{ miles}}\)
3. Length of the other leg of the triangle: \(\boxed{102.06 \text{ mm}}\)
4. Distance from the base of the house for the ladder: \(\boxed{9.95 \text{ feet}}\)