Pythagoras Theorem
The foundation of right-angled triangle calculations
The Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
\[c^2 = a^2 + b^2\]
Where c is the hypotenuse (longest side, opposite the right angle)
Key Terms
Hypotenuse: The longest side of a right-angled triangle, always opposite the 90° angle.
Legs/Catheti: The two shorter sides that form the right angle.
Right angle: An angle of exactly 90°, often marked with a small square.
Using the Theorem
Example 1: Finding the Hypotenuse
A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
\[c^2 = a^2 + b^2\]
\[c^2 = 6^2 + 8^2 = 36 + 64 = 100\]
\[c = \sqrt{100} = 10 \text{ cm}\]
Example 2: Finding a Leg
A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.
\[a^2 = c^2 - b^2\]
\[a^2 = 13^2 - 5^2 = 169 - 25 = 144\]
\[a = \sqrt{144} = 12 \text{ cm}\]
Interactive Pythagoras Calculator
Calculate Missing Sides
\(c^2 = 3^2 + 4^2 = 9 + 16 = 25\)
\(c = \sqrt{25} = 5\)
Pythagorean Triples
What are Pythagorean Triples?
A Pythagorean triple consists of three positive integers a, b, and c that satisfy \(a^2 + b^2 = c^2\).
Memorizing common triples can save time in exams!
Note: 6, 8, 10 and 9, 12, 15 are multiples of the 3, 4, 5 triple!
Real-World Applications
Ladder Problems
Finding how far a ladder reaches up a wall, or how far from the wall to place it
Distance Problems
Finding the shortest distance between two points (diagonal distance)
Construction
Ensuring corners are exactly 90° (the 3-4-5 method)
Navigation
Calculating distances traveled when changing direction
Example 3: Ladder Problem
A 10 m ladder leans against a wall with its base 6 m from the wall. How high up the wall does the ladder reach?
Ladder = hypotenuse = 10 m, Base = 6 m, Height = ?
\[h^2 = 10^2 - 6^2 = 100 - 36 = 64\]
\[h = \sqrt{64} = 8 \text{ m}\]
Example 4: Distance Problem
A ship sails 12 km North and then 9 km East. How far is it from its starting point?
The path forms a right-angled triangle. The direct distance is the hypotenuse.
\[d^2 = 12^2 + 9^2 = 144 + 81 = 225\]
\[d = \sqrt{225} = 15 \text{ km}\]
Testing for Right Angles
The Converse of Pythagoras
If \(a^2 + b^2 = c^2\) for the three sides of a triangle, then the triangle is right-angled.
If \(a^2 + b^2 > c^2\): The triangle is acute (all angles < 90°)
If \(a^2 + b^2 < c^2\): The triangle is obtuse (one angle > 90°)
Practice Problems
Question 1: Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.
Show Solution
\(c^2 = 5^2 + 12^2 = 25 + 144 = 169\)
\(c = \sqrt{169} = 13\) cm
(This is the 5, 12, 13 Pythagorean triple!)
Question 2: A rectangle is 8 m long and 6 m wide. Find the length of its diagonal.
Show Solution
The diagonal forms a right triangle with the length and width.
\(d^2 = 8^2 + 6^2 = 64 + 36 = 100\)
\(d = \sqrt{100} = 10\) m
Question 3: Is a triangle with sides 7, 24, and 25 a right-angled triangle?
Show Solution
Check if \(a^2 + b^2 = c^2\):
\(7^2 + 24^2 = 49 + 576 = 625\)
\(25^2 = 625\)
Yes! \(625 = 625\), so it IS a right-angled triangle.
Question 4: A football field is 100 m long and 64 m wide. A player runs diagonally from one corner to the opposite corner. How far does he run?
Show Solution
\(d^2 = 100^2 + 64^2 = 10000 + 4096 = 14096\)
\(d = \sqrt{14096} \approx 118.7\) m
CSEC Exam Tips
- Always identify the hypotenuse first (longest side, opposite 90°)
- Memorize common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
- Draw a diagram for word problems
- Check your answer: the hypotenuse must be the longest side
- Round appropriately as instructed (usually 1-3 decimal places)