Pythagoras Theorem - Grade 9 Math
Key Concepts
1. Definition of Pythagoras Theorem
Pythagoras Theorem 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, it is expressed as:
\(a^2 + b^2 = c^2\)
where \(a\) and \(b\) are the lengths of the legs of the triangle, and \(c\) is the length of the hypotenuse.
2. Application of Pythagoras Theorem
The theorem is used to find the length of any side of a right-angled triangle when the lengths of the other two sides are known. It is also used in various real-world applications, such as construction, navigation, and physics.
3. Special Triangles
Certain right-angled triangles have specific side lengths that are commonly used in problems. For example, the 3-4-5 triangle is a special right-angled triangle where the sides are in the ratio 3:4:5. This ratio is often used to simplify calculations.
Detailed Explanation
Example 1: Using Pythagoras Theorem to Find the Hypotenuse
Problem: Find the length of the hypotenuse of a right-angled triangle with legs of lengths 6 cm and 8 cm.
Solution:
Using the Pythagoras Theorem:
\(a = 6\), \(b = 8\)
\(c^2 = a^2 + b^2\)
\(c^2 = 6^2 + 8^2\)
\(c^2 = 36 + 64\)
\(c^2 = 100\)
\(c = \sqrt{100}\)
\(c = 10\)
The length of the hypotenuse is 10 cm.
Example 2: Using Pythagoras Theorem to Find a Leg
Problem: Find the length of one leg of a right-angled triangle with a hypotenuse of 13 cm and the other leg of 5 cm.
Solution:
Using the Pythagoras Theorem:
\(c = 13\), \(a = 5\)
\(c^2 = a^2 + b^2\)
\(13^2 = 5^2 + b^2\)
\(169 = 25 + b^2\)
\(b^2 = 169 - 25\)
\(b^2 = 144\)
\(b = \sqrt{144}\)
\(b = 12\)
The length of the other leg is 12 cm.
Example 3: Special Triangles - 3-4-5 Triangle
Problem: Verify the Pythagoras Theorem for a 3-4-5 triangle.
Solution:
For a 3-4-5 triangle:
\(a = 3\), \(b = 4\), \(c = 5\)
Using the Pythagoras Theorem:
\(c^2 = a^2 + b^2\)
\(5^2 = 3^2 + 4^2\)
\(25 = 9 + 16\)
\(25 = 25\)
The equation holds true, verifying the Pythagoras Theorem for a 3-4-5 triangle.
Analogies for Clarity
Pythagoras Theorem as a Puzzle
Think of the Pythagoras Theorem as a puzzle where you need to fit three squares together. The largest square (hypotenuse) is made by combining the areas of the two smaller squares (legs). Solving the puzzle means finding the correct side lengths that fit perfectly together.
Pythagoras Theorem as a Bridge
Consider the theorem as a bridge that connects the lengths of the sides of a right-angled triangle. The bridge ensures that the relationship between the sides is always balanced, allowing you to cross from knowing two sides to finding the third.