Pythagoras Secret Slides
Geometry Foundations Grades 6–7
The Secret of the
Right Triangle
Discovering how squares build the famous rule: \(a^2 + b^2 = c^2\)
Triangle Anatomy
Square Area Proof
The 3-4-5 Triangle
Step 1: The Basics
Anatomy of a Right Triangle
01 / 07
Leg a Leg b Hypotenuse (c)
The 90° corner always points at the hypotenuse!
1 The Legs (\(a\) and \(b\))
The two shorter sides that meet to form the crisp 90° right angle (the "L" corner).
2 The Hypotenuse (\(c\))
The longest side of all. It sits directly across the triangle from the 90° angle.
Rule to remember: The hypotenuse is never one of the sides touching the 90° square corner!
Step 2: The Visual Proof
What If We Build Squares on Each Side?
02 / 07
9 3 × 3 = 9 16 4 × 4 = 16 25 5 × 5 = 25
Square on Leg 3:
\(3 \times 3 = 9\) tiles
Square on Leg 4:
\(4 \times 4 = 16\) tiles
Add them together!
9 + 16 = 25
It perfectly matches the hypotenuse square!
The two smaller square areas always combine to equal the giant square area!
Step 3: The Big Formula
The Pythagorean Theorem
03 / 07
In every right triangle:
\(a^2\) + \(b^2\) = \(c^2\)
(leg squared) + (leg squared) = (hypotenuse squared)
a
First Leg
Square its length: multiply it by itself (\(a \times a\)).
b
Second Leg
Square its length: multiply it by itself (\(b \times b\)).
c
Hypotenuse
The sum gives \(c^2\). Take the square root to get \(c\)!
Crucial Condition: This theorem only works if the triangle has a 90-degree right angle!
Step 4: Worked Example
Solving for the Missing Side
04 / 07
a = 3 b = 4 c = ?
Find the length of hypotenuse \(c\)
1 Write the formula
\(a^2 + b^2 = c^2\)
2 Plug in legs \(3\) and \(4\)
\(3^2 + 4^2 = c^2\)
3 Square and add
\(9 + 16 = 25 = c^2\)
4 Take the square root
\(c = \sqrt{25} = 5\)
Answer Check: Since \(5 > 4\) and \(5 > 3\), the hypotenuse is indeed the longest side!
Step 5: Guided Practice
Your Turn: Find Hypotenuse \(c\)!
05 / 07
a = 6 b = 8 c = ?
Leg lengths: 6 and 8
Question 1: What is \(6^2\) and \(8^2\)?
\(6 \times 6 = 36\) | \(8 \times 8 = 64\)
Question 2: What is the sum \(36 + 64\)?
\(36 + 64 = 100 = c^2\)
Final Question: What times itself equals 100?
\(\sqrt{100} = \) 10
So hypotenuse \(c = 10\)!
Cool Math Pattern: Notice that 6-8-10 is just the 3-4-5 triangle multiplied by 2!
Step 6: Common Trap
Triangles Can Be Turned in Any Direction!
06 / 07
Orientation A c is here!
Hypotenuse is tilted at the bottom.
Orientation B c is here!
Hypotenuse is the slanted ramp.
Orientation C c is here!
Hypotenuse stretches across the top.
The Secret Trick: Put your pencil tip on the 90° square box and draw an arrow straight out into the triangle. It will always hit side \(c\)!
Review & Wrap-Up
Three Golden Rules to Remember
07 / 07
1. Right Triangles Only
The theorem only works when there is an exact 90° corner.
Check for the square box!
2. The Magic Equation
\(a^2 + b^2 = c^2\)
Square the two legs, add them, and find the square root.
Legs squared = Hypotenuse squared
3. \(c\) is the Champion
Side \(c\) is always the longest side, sitting directly across from the 90° angle.
Longest side is always \(c\)!
Ready to test your triangle superpowers? Grab your grid paper and let's go! Pythagoras Mastered
