Pythagorean Theorem Word Problems Worksheets - Free Printable

Let's solve each problem step by step using the Pythagorean Theorem, which states:

$$
a^2 + b^2 = c^2
$$

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

---

Problem 1:

Dan secures a ladder firmly to the wall to dust a photo frame mounted on the wall. The foot of the ladder is placed 6 feet from the wall. If the top of the ladder rests 8 feet up the wall, what is the length of the ladder?

#### Solution:
- The ladder forms the hypotenuse of a right triangle.
- The distance from the wall to the foot of the ladder is one leg (\( a = 6 \) feet).
- The height up the wall where the ladder rests is the other leg (\( b = 8 \) feet).
- We need to find the length of the ladder (\( c \)).

Using the Pythagorean Theorem:
$$
a^2 + b^2 = c^2
$$
Substitute the given values:
$$
6^2 + 8^2 = c^2
$$
$$
36 + 64 = c^2
$$
$$
100 = c^2
$$
Take the square root of both sides:
$$
c = \sqrt{100} = 10
$$

Thus, the length of the ladder is:
$$
\boxed{10}
$$

---

Problem 2:

If the playing surface of a carom board is a square of length 28 inches, what is the diagonal length of the playing surface?

#### Solution:
- The carom board is a square with side length \( s = 28 \) inches.
- The diagonal of a square divides it into two congruent right triangles.
- In each triangle, the diagonal is the hypotenuse, and the sides of the square are the legs.

Using the Pythagorean Theorem:
$$
s^2 + s^2 = d^2
$$
where \( d \) is the diagonal. Substitute \( s = 28 \):
$$
28^2 + 28^2 = d^2
$$
$$
784 + 784 = d^2
$$
$$
1568 = d^2
$$
Take the square root of both sides:
$$
d = \sqrt{1568}
$$
Simplify:
$$
d = \sqrt{1568} = \sqrt{16 \times 98} = 4\sqrt{98} = 4\sqrt{49 \times 2} = 4 \times 7\sqrt{2} = 28\sqrt{2}
$$
Approximate \( \sqrt{2} \approx 1.414 \):
$$
d \approx 28 \times 1.414 = 39.592
$$
Round to the nearest tenth:
$$
d \approx 39.6
$$

Thus, the diagonal length is:
$$
\boxed{39.6}
$$

---

Problem 3:

Carlos flies his drone 5 feet above the ground. He rotates the drone to the right. So far, the drone has flown a total of 12 feet. How far is the drone from the start point?

#### Solution:
- The drone flies horizontally for a distance of 12 feet.
- The drone is also 5 feet above the ground, forming a right triangle.
- The horizontal distance is one leg (\( a = 12 \) feet).
- The vertical distance is the other leg (\( b = 5 \) feet).
- We need to find the straight-line distance from the start point to the drone (\( c \)).

Using the Pythagorean Theorem:
$$
a^2 + b^2 = c^2
$$
Substitute the given values:
$$
12^2 + 5^2 = c^2
$$
$$
144 + 25 = c^2
$$
$$
169 = c^2
$$
Take the square root of both sides:
$$
c = \sqrt{169} = 13
$$

Thus, the distance from the start point is:
$$
\boxed{13}
$$

---

Problem 4:

A ship is on its maiden voyage to the mystery island. According to the map, it has to travel 99 miles to the north and then 20 miles to the west to reach the island. Find the shortest route that leads to the mystery island.

#### Solution:
- The ship travels 99 miles north and then 20 miles west, forming a right triangle.
- The northward distance is one leg (\( a = 99 \) miles).
- The westward distance is the other leg (\( b = 20 \) miles).
- We need to find the shortest route, which is the hypotenuse (\( c \)).

Using the Pythagorean Theorem:
$$
a^2 + b^2 = c^2
$$
Substitute the given values:
$$
99^2 + 20^2 = c^2
$$
$$
9801 + 400 = c^2
$$
$$
10201 = c^2
$$
Take the square root of both sides:
$$
c = \sqrt{10201} = 101
$$

Thus, the shortest route is:
$$
\boxed{101}
$$

---

Problem 5:

For her boat model, Dahlia cut a piece of fabric in the shape of a right triangle. If the length of the longest side was 8 inches and one of the other two sides measured 4 inches, what was the length of the other side?

#### Solution:
- The longest side of a right triangle is the hypotenuse (\( c = 8 \) inches).
- One of the other sides is given as 4 inches (\( a = 4 \) inches).
- We need to find the length of the remaining side (\( b \)).

Using the Pythagorean Theorem:
$$
a^2 + b^2 = c^2
$$
Substitute the given values:
$$
4^2 + b^2 = 8^2
$$
$$
16 + b^2 = 64
$$
Isolate \( b^2 \):
$$
b^2 = 64 - 16
$$
$$
b^2 = 48
$$
Take the square root of both sides:
$$
b = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}
$$
Approximate \( \sqrt{3} \approx 1.732 \):
$$
b \approx 4 \times 1.732 = 6.928
$$
Round to the nearest tenth:
$$
b \approx 6.9
$$

Thus, the length of the other side is:
$$
\boxed{6.9}
$$

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Final Answers:

1. \( \boxed{10} \)
2. \( \boxed{39.6} \)
3. \( \boxed{13} \)
4. \( \boxed{101} \)
5. \( \boxed{6.9} \)