Understanding Pythagorean Theorem
Introduction and Warm‑Up
In this lesson, you’ll explore the Pythagorean Theorem, learn to prove it, and apply it to real‑world and coordinate‑geometry problems. Before we start, recall what a right triangle is: a triangle with one 90° angle.[br]
[i]Which of these is true about a right triangle?[/i][br]
Building a Draggable Right Triangle with Squares
Click and drag the red point. How does it move? Do we always have a right triangle? Why?
[i]Which equation correctly represents the relationship between the sides of a right triangle?[br][/i][br][br]
Proving the Pythagorean Theorem
[list][*][i]Let’s prove the theorem by rearranging shapes. Click on any point and move it and observe how the two smaller squares perfectly fill the largest square.[/i][br][/*][/list]
[list][*][i]What did you notice during the animation? Why do the squares’ areas remain the same even when rearranged?[/i][br][/*][/list]
Testing the Converse
If [math]a^2+b^2=c^2[/math] [i]then the triangle must be right‑angled. Try it![/i]
Common Misconceptions
[i]The Pythagorean Theorem works for all triangles.[/i]
[i]The hypotenuse is always opposite the right angle.[/i]
[i]To find a missing leg, subtract the squared lengths: b² = c² − a².[/i]
[i]Side lengths can be negative.[/i]
Find the missing side
[i]Given a right triangle with legs a = 5 and hypotenuse c = 13, what is the length of the other leg b?[/i][br]
Real-Life Problem
[i][size=100][size=150]A firefighter places a 40‑ft ladder against a wall. The foot of the ladder is 9 ft from the wall. [/size][/size][/i][br]
[i]How high up the wall does the ladder reach?[/i]
Exit Ticket/ Reflection
[list=1][*][i]“Why does the Pythagorean Theorem apply only to right triangles?”[/i][br][/*][*][i]“Describe one misconception you had and how today’s exploration changed your understanding.”[/i][/*][/list]