Pythagorean Theorem Worksheets - Math Monks - Free Printable

We are given a Pythagorean Theorem Word Problems Worksheet. The Pythagorean Theorem states:

> In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This is written as:
> a² + b² = c²

Where:
- `a` and `b` are the legs (perpendicular sides),
- `c` is the hypotenuse (longest side, opposite the right angle).

Let’s solve each problem one by one.

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Problem 1:

> The house is 12 feet tall. The man is 5 feet from the house. Solve for the missing length ‘c’.

This forms a right triangle:
- Vertical leg (height of house) = 12 ft
- Horizontal leg (distance from man to house) = 5 ft
- Hypotenuse (line from man’s eyes to top of house) = c

Using Pythagorean Theorem:

c² = 12² + 5²
c² = 144 + 25
c² = 169
c = √169 = 13

✔ Answer: 13 feet

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Problem 2:

> A 12 foot ladder is leaning against a wall at a distance of 3 feet. Find the distance from the ground to the end of the ladder where it touches the wall.

This is also a right triangle:
- Ladder = hypotenuse = 12 ft
- Distance from wall = base = 3 ft
- Height on wall = unknown leg = let's call it `h`

So:

h² + 3² = 12²
h² + 9 = 144
h² = 144 - 9 = 135
h = √135

Simplify √135:
√135 = √(9 × 15) = 3√15 ≈ 11.62 feet (if decimal needed)

But since the worksheet likely expects exact form or simplified radical:

✔ Answer: √135 feet or 3√15 feet

*(Note: If decimal approximation is required, it’s about 11.6 ft)*

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Problem 3:

> A 6 foot tall tree casts a shadow that is 8 feet long. What is the distance from the top of the tree to the top of its shadow?

This forms a right triangle:
- Tree height = 6 ft (vertical leg)
- Shadow length = 8 ft (horizontal leg)
- Distance from top of tree to top of shadow = hypotenuse = `d`

d² = 6² + 8²
d² = 36 + 64 = 100
d = √100 = 10

✔ Answer: 10 feet

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Problem 4:

> A flag pole of height 20 feet casts a shadow of 5 feet. What is the distance from the top of the pole to the end of the shadow?

Again, right triangle:
- Pole height = 20 ft (vertical leg)
- Shadow = 5 ft (horizontal leg)
- Distance from top of pole to end of shadow = hypotenuse = `d`

d² = 20² + 5²
d² = 400 + 25 = 425
d = √425

Simplify √425:
√425 = √(25 × 17) = 5√17

✔ Answer: √425 feet or 5√17 feet

*(Approximately 20.62 feet if decimal needed)*

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## ✔ Final Answers:

1. 13 feet
2. √135 feet or 3√15 feet
3. 10 feet
4. √425 feet or 5√17 feet

These answers use the Pythagorean Theorem correctly for each real-world scenario forming a right triangle.