Problem Analysis:
The worksheet involves applying the
Pythagorean Theorem to solve real-world problems. The Pythagorean Theorem states:
\[
a^2 + b^2 = c^2
\]
where \( c \) is the hypotenuse (the longest side) of a right triangle, and \( a \) and \( b \) are the other two sides.
We will solve each problem step by step.
---
Problem 1:
-
Given:
- Height of the house (\( a \)) = 12 feet
- Distance from the man to the house (\( b \)) = 5 feet
- Missing length (\( c \)) is the hypotenuse.
-
Solution:
Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the given values:
\[
12^2 + 5^2 = c^2
\]
Calculate the squares:
\[
144 + 25 = c^2
\]
Add the values:
\[
169 = c^2
\]
Take the square root of both sides:
\[
c = \sqrt{169} = 13
\]
-
Answer:
\[
\boxed{13}
\]
---
Problem 2:
-
Given:
- Length of the ladder (\( c \)) = 12 feet
- Distance from the wall to the base of the ladder (\( b \)) = 3 feet
- Missing length (\( a \)) is the height from the ground to where the ladder touches the wall.
-
Solution:
Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the given values:
\[
a^2 + 3^2 = 12^2
\]
Calculate the squares:
\[
a^2 + 9 = 144
\]
Subtract 9 from both sides:
\[
a^2 = 135
\]
Take the square root of both sides:
\[
a = \sqrt{135} = 3\sqrt{15}
\]
-
Answer:
\[
\boxed{3\sqrt{15}}
\]
---
Problem 3:
-
Given:
- Height of the tree (\( a \)) = 6 feet
- Length of the shadow (\( b \)) = 8 feet
- Missing length (\( c \)) is the distance from the top of the tree to the top of its shadow.
-
Solution:
Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the given values:
\[
6^2 + 8^2 = c^2
\]
Calculate the squares:
\[
36 + 64 = c^2
\]
Add the values:
\[
100 = c^2
\]
Take the square root of both sides:
\[
c = \sqrt{100} = 10
\]
-
Answer:
\[
\boxed{10}
\]
---
Problem 4:
-
Given:
- Height of the flag pole (\( a \)) = 20 feet
- Length of the shadow (\( b \)) = 5 feet
- Missing length (\( c \)) is the distance from the top of the pole to the end of the shadow.
-
Solution:
Using the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]
Substitute the given values:
\[
20^2 + 5^2 = c^2
\]
Calculate the squares:
\[
400 + 25 = c^2
\]
Add the values:
\[
425 = c^2
\]
Take the square root of both sides:
\[
c = \sqrt{425} = 5\sqrt{17}
\]
-
Answer:
\[
\boxed{5\sqrt{17}}
\]
---
Final Answers:
1. \(\boxed{13}\)
2. \(\boxed{3\sqrt{15}}\)
3. \(\boxed{10}\)
4. \(\boxed{5\sqrt{17}}\)
Parent Tip: Review the logic above to help your child master the concept of pythagoras word problems worksheet.
