Let's solve each of these
Pythagorean Theorem word problems step by step. The Pythagorean Theorem 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, opposite the right angle).
---
1) Dan secures a ladder firmly to the wall...
- The foot of the ladder is 6 feet from the wall.
- The top of the ladder rests 8 feet up the wall.
- We need to find the length of the ladder (the hypotenuse).
Let’s assign:
- $ a = 6 $ ft (horizontal leg)
- $ b = 8 $ ft (vertical leg)
- $ c = ? $ (ladder length)
$$
c^2 = 6^2 + 8^2 = 36 + 64 = 100
$$
$$
c = \sqrt{100} = 10
$$
✔ Answer: 10.0 feet
---
2) Diagonal of a square carom board with side 28 inches
A square has equal sides. The diagonal forms a right triangle with two sides of the square.
So:
- $ a = 28 $
- $ b = 28 $
- $ c = ? $
$$
c^2 = 28^2 + 28^2 = 784 + 784 = 1568
$$
$$
c = \sqrt{1568} \approx 39.6
$$
✔ Answer: 39.6 inches
---
3) Carlos flies his drone 5 feet above ground, then rotates it right and flies 12 feet
This describes a right triangle:
- Vertical leg: 5 ft (height)
- Horizontal leg: 12 ft (distance flown to the right)
- Hypotenuse: distance from start point
$$
c^2 = 5^2 + 12^2 = 25 + 144 = 169
$$
$$
c = \sqrt{169} = 13
$$
✔ Answer: 13.0 feet
---
4) Ship travels 9 miles north, then 20 miles west — find shortest route
The ship's path forms a right triangle:
- North: 9 miles (one leg)
- West: 20 miles (other leg)
- Shortest route = straight-line distance (hypotenuse)
$$
c^2 = 9^2 + 20^2 = 81 + 400 = 481
$$
$$
c = \sqrt{481} \approx 21.9
$$
✔ Answer: 21.9 miles
---
5) Dahlia cuts fabric in shape of right triangle: longest side 8 inches, one other side 4 inches — find third side
Longest side = hypotenuse = 8 inches
One leg = 4 inches
Find the other leg ($ x $):
$$
x^2 + 4^2 = 8^2
$$
$$
x^2 + 16 = 64
$$
$$
x^2 = 48
$$
$$
x = \sqrt{48} \approx 6.9
$$
✔ Answer: 6.9 inches
---
✔ Final Answers (rounded to nearest tenth):
1)
10.0 feet
2)
39.6 inches
3)
13.0 feet
4)
21.9 miles
5)
6.9 inches
Let me know if you'd like a visual explanation or diagram for any!
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem problems worked out.
