Pythagoras and Triangles Problem Solving
Year 10 Mathematics NSW Curriculum Using Pythagoras' Theorem to Solve Real-World Problems
Who Was Pythagoras?
Ancient Greek philosopher and mathematician (c. 570-495 BCE) Founded the Pythagorean school in ancient Greece Believed numbers were the key to understanding the universe His theorem is one of the most famous in mathematics
The Pythagorean Theorem
In a right-angled triangle: a² + b² = c² a and b are the shorter sides (legs) c is the longest side (hypotenuse) Only works for right-angled triangles
Quick Check: Identify the Hypotenuse
Look at the triangles on the board Point to the hypotenuse in each triangle Remember: it's always opposite the right angle It's always the longest side
Finding the Hypotenuse
When you know both legs (a and b) Use: c = √(a² + b²) Example: If a = 3 and b = 4 Then c = √(3² + 4²) = √(9 + 16) = √25 = 5
Finding a Leg
When you know the hypotenuse (c) and one leg Rearrange to: a = √(c² - b²) or b = √(c² - a²) Example: If c = 13 and b = 5 Then a = √(13² - 5²) = √(169 - 25) = √144 = 12
Practice Problem
A ladder is 10 meters long It leans against a wall The base is 6 meters from the wall How high up the wall does it reach?
Area of Right-Angled Triangles
Area = ½ × base × height For right triangles, use the two legs as base and height The legs are perpendicular to each other Example: legs of 6 cm and 8 cm give area = ½ × 6 × 8 = 24 cm²
Pythagoras vs Area Calculations
{"left":"Pythagoras finds missing side lengths\nUses a² + b² = c²\nInvolves square roots\nGives measurements in linear units (cm, m)","right":"Area finds space inside triangle\nUses Area = ½ × base × height\nInvolves multiplication and division\nGives measurements in square units (cm², m²)"}
Real-World Problem Solving
Work in pairs Choose a problem from the worksheet Decide: Do you need Pythagoras or area? Show all working clearly
Surveying Applications
Surveyors use Pythagoras to measure distances Traverse offset surveys create right triangles Measure accessible distances to find inaccessible ones Essential for mapping and construction
Challenge Problem
A rectangular field is 80m by 60m A path goes diagonally across it What is the length of the path? What is the area of the field?
2 more slides available after you open the deck.
Download all 14 slides