Problem Description:
The task involves solving problems related to
Pythagoras' Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as:
\[
c^2 = a^2 + b^2
\]
where:
- \( c \) is the length of the hypotenuse,
- \( a \) and \( b \) are the lengths of the other two sides.
The provided image contains two examples:
1.
Example 1: Longest Side
2.
Example 2: Shorter Side
We will solve both examples step by step.
---
Example 1: Longest Side
#### Given:
- One side (\( a \)) = 3 cm
- Another side (\( b \)) = 4 cm
- Hypotenuse (\( c \)) = ?
#### Steps to Solve:
1.
Write down Pythagoras' Theorem:
\[
c^2 = a^2 + b^2
\]
2.
Substitute the given values:
\[
c^2 = 3^2 + 4^2
\]
3.
Calculate the squares:
\[
3^2 = 9, \quad 4^2 = 16
\]
\[
c^2 = 9 + 16
\]
4.
Add the values:
\[
c^2 = 25
\]
5.
Take the square root to find \( c \):
\[
c = \sqrt{25} = 5
\]
#### Final Answer for Example 1:
\[
\boxed{5 \text{ cm}}
\]
---
Example 2: Shorter Side
#### Given:
- Hypotenuse (\( c \)) = 5 cm
- One side (\( a \)) = 3 cm
- Another side (\( b \)) = ?
#### Steps to Solve:
1.
Write down Pythagoras' Theorem:
\[
c^2 = a^2 + b^2
\]
2.
Substitute the given values:
\[
5^2 = 3^2 + b^2
\]
3.
Calculate the squares:
\[
5^2 = 25, \quad 3^2 = 9
\]
\[
25 = 9 + b^2
\]
4.
Isolate \( b^2 \):
\[
b^2 = 25 - 9
\]
\[
b^2 = 16
\]
5.
Take the square root to find \( b \):
\[
b = \sqrt{16} = 4
\]
#### Final Answer for Example 2:
\[
\boxed{4 \text{ cm}}
\]
---
Summary:
-
Example 1 (Longest Side): The hypotenuse is \( \boxed{5 \text{ cm}} \).
-
Example 2 (Shorter Side): The missing side is \( \boxed{4 \text{ cm}} \).
These solutions demonstrate how to apply Pythagoras' Theorem to find either the hypotenuse or one of the shorter sides of a right-angled triangle.
Parent Tip: Review the logic above to help your child master the concept of triangle trigonometry worksheet.
