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,
- $ c $ is the hypotenuse (the side opposite the right angle).
---
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): $ a = 12 $ ft
- Horizontal leg (distance from man to house): $ b = 5 $ ft
- Hypotenuse $ c $ = distance from man to top of house
$$
c^2 = a^2 + b^2 = 12^2 + 5^2 = 144 + 25 = 169
$$
$$
c = \sqrt{169} = 13
$$
✔ Answer: $ \boxed{13} $ feet
---
Problem 2:
> A 12 feet 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.
Right triangle:
- Hypotenuse (ladder): $ c = 12 $ ft
- Base (distance from wall): $ a = 3 $ ft
- Height (what we're solving for): $ b = ? $
$$
a^2 + b^2 = c^2
\Rightarrow 3^2 + b^2 = 12^2
\Rightarrow 9 + b^2 = 144
\Rightarrow b^2 = 135
\Rightarrow b = \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}
$$
Approximate value: $ \sqrt{135} \approx 11.62 $ ft
✔ Answer: $ \boxed{3\sqrt{15}} $ ft or approximately $ \boxed{11.62} $ feet
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Problem 3:
> A 6 feet 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?
Right triangle:
- Height of tree: $ a = 6 $ ft
- Length of shadow: $ b = 8 $ ft
- Hypotenuse $ c $ = distance from top of tree to tip of shadow
$$
c^2 = 6^2 + 8^2 = 36 + 64 = 100
\Rightarrow c = \sqrt{100} = 10
$$
✔ Answer: $ \boxed{10} $ feet
---
Problem 4:
> A flagpole 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?
Right triangle:
- Height of flagpole: $ a = 20 $ ft
- Shadow length: $ b = 5 $ ft
- Hypotenuse $ c $ = distance from top of pole to end of shadow
$$
c^2 = 20^2 + 5^2 = 400 + 25 = 425
\Rightarrow c = \sqrt{425} = \sqrt{25 \times 17} = 5\sqrt{17}
$$
Approximate: $ \sqrt{425} \approx 20.62 $ ft
✔ Answer: $ \boxed{5\sqrt{17}} $ ft or approximately $ \boxed{20.62} $ feet
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✔ Final Answers Summary:
1. $ \boxed{13} $ ft
2. $ \boxed{3\sqrt{15}} $ ft ≈ $ \boxed{11.62} $ ft
3. $ \boxed{10} $ ft
4. $ \boxed{5\sqrt{17}} $ ft ≈ $ \boxed{20.62} $ ft
Let me know if you'd like these answers written in a specific format!
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem worksheet word problems.
