Pythagoras’ Theorem: Your Guide to Right-Angled Triangles!
Hey everyone! Welcome to your study notes for Pythagoras’ Theorem. This sounds like a big, fancy name, but don't worry! It's actually a super useful and quite simple rule about a special type of triangle. In these notes, we'll unlock its secrets together.
You'll learn what the theorem is, how to use it, and why it's a super-power for solving real-world problems, from building a shelf to playing video games. Let's get started!
First Things First: Meet the Right-Angled Triangle
Before we can use the theorem, we need to know our main character: the right-angled triangle. It's any triangle that has one perfect corner, like the corner of a square or a book. This special angle is 90 degrees.
The Parts of a Right-Angled Triangle
Every right-angled triangle has three sides with special names. Getting these right is the most important first step!
The Hypotenuse (we'll call it 'c'): This is the superstar of the triangle. It's ALWAYS the longest side, and it's ALWAYS opposite the right angle. Think of it as the 'sloping' side.
The Legs (we'll call them 'a' and 'b'): These are the other two sides. They are the ones that meet to form the 90-degree right angle. It doesn't matter which one you call 'a' and which one you call 'b'.
Imagine a triangle with a square symbol in one corner. That's the right angle. The side that doesn't touch that corner is the hypotenuse!
Quick Review Box
Right-Angled Triangle: A triangle with one 90° angle.
Hypotenuse (c): The longest side, opposite the right angle.
Legs (a and b): The two shorter sides that form the right angle.
Key Takeaway
If you can correctly spot the hypotenuse, you're already halfway to mastering this topic. Great job!
The Main Event: What is Pythagoras' Theorem?
Pythagoras' Theorem is a special relationship between the sides of a right-angled triangle. It says:
"If you make a square on each of the two shorter sides (the legs), their areas will add up to the exact same area as the square on the longest side (the hypotenuse)."
That's cool, but it's much easier to use as a formula!
The Famous Formula
The theorem is written as this simple and powerful equation:
$$ a^2 + b^2 = c^2 $$Where:
a and b are the lengths of the legs.
c is the length of the hypotenuse.
This formula lets you find the length of a missing side if you know the other two. Let's see how!
Example 1: Finding the Hypotenuse (c)
Imagine a triangle with legs of length 3 cm and 4 cm. We want to find the hypotenuse.
Step-by-Step Guide:
Identify your sides:
The legs are a = 3 and b = 4. We are looking for c.Write down the formula:
$$ a^2 + b^2 = c^2 $$Substitute the numbers you know:
$$ 3^2 + 4^2 = c^2 $$Calculate the squares:
Remember, $$3^2$$ means 3 × 3.
$$ 9 + 16 = c^2 $$Add the numbers:
$$ 25 = c^2 $$Find the square root:
To get 'c' by itself, we need to do the opposite of squaring, which is finding the square root ($$\sqrt{...}$$).
$$ c = \sqrt{25} $$
$$ c = 5 $$
Answer: The hypotenuse is 5 cm long!
Example 2: Finding a Leg (a or b)
Now, imagine a triangle where the hypotenuse is 13 cm and one leg is 12 cm. Let's find the other leg.
Don't worry, the process is very similar!
Step-by-Step Guide:
Identify your sides:
The hypotenuse c = 13. One leg is b = 12. We are looking for a.Write down the formula:
$$ a^2 + b^2 = c^2 $$Substitute the numbers you know:
$$ a^2 + 12^2 = 13^2 $$Calculate the squares:
$$ a^2 + 144 = 169 $$Rearrange the formula to find $$a^2$$:
We need to get $$a^2$$ alone, so we subtract 144 from both sides.
$$ a^2 = 169 - 144 $$Subtract the numbers:
$$ a^2 = 25 $$Find the square root:
$$ a = \sqrt{25} $$
$$ a = 5 $$
Answer: The missing leg is 5 cm long!
Common Mistakes to Avoid!
Mixing up 'c': Always make sure 'c' is the hypotenuse (the longest side). The formula is NOT $$a^2 + c^2 = b^2$$!
Forgetting the final step: A common mistake is stopping at $$c^2 = 25$$. You must find the square root to get the final answer for 'c'.
Adding when you should subtract: When you are looking for a shorter side (a or b), remember to subtract the squares, like in Example 2.
Key Takeaway
The formula $$a^2 + b^2 = c^2$$ is your key! If you're finding the longest side (c), you add. If you're finding a shorter side (a or b), you subtract.
The Other Way Around: The Converse of the Theorem
So, we know that if a triangle has a right angle, then $$a^2 + b^2 = c^2$$.
The converse is just the reverse of that statement:
"If a triangle's sides fit the formula $$a^2 + b^2 = c^2$$, then it must be a right-angled triangle."
We use the converse to test if a triangle has a 90° angle.
How to Use the Converse
A triangle has sides of 8 cm, 15 cm, and 17 cm. Is it a right-angled triangle?
Step-by-Step Test:
Identify the longest side. This MUST be your 'c'.
Here, c = 17. So, a = 8 and b = 15.Calculate $$a^2 + b^2$$ separately.
$$ 8^2 + 15^2 = 64 + 225 = 289 $$Now, calculate $$c^2$$ separately.
$$ 17^2 = 289 $$Compare your results.
Does $$a^2 + b^2 = c^2$$? Yes! 289 is equal to 289.
Conclusion: Because the sides fit the formula, this is a right-angled triangle!
What if the sides were 5, 6, and 7?
Longest side c = 7.
$$a^2 + b^2 = 5^2 + 6^2 = 25 + 36 = 61$$
$$c^2 = 7^2 = 49$$
61 is NOT equal to 49, so this is not a right-angled triangle.
Key Takeaway
The converse helps you become a "triangle detective." If $$a^2 + b^2$$ equals $$c^2$$, you've found a right angle!
Pythagoras in Action! Solving Real Problems
This is where the magic happens. Pythagoras' Theorem is used all the time in the real world.
Top Tip: Always draw a simple picture for word problems. It helps you see the triangle!
Problem Example: The Leaning Ladder
A 10-metre ladder is leaning against a wall. The bottom of the ladder is 6 metres away from the base of the wall. How high up the wall does the ladder reach?
Solving the Problem:
Draw a picture. You'll see the ladder, the wall, and the ground form a right-angled triangle.
The ladder is the sloping side, so it's the hypotenuse (c = 10).
The ground is one leg (b = 6).
The wall is the other leg, which we need to find (a = ?).
Choose your formula. We are finding a leg, so we will need to subtract.
$$ a^2 + b^2 = c^2 $$Substitute and solve:
$$ a^2 + 6^2 = 10^2 $$
$$ a^2 + 36 = 100 $$
$$ a^2 = 100 - 36 $$
$$ a^2 = 64 $$
$$ a = \sqrt{64} $$
$$ a = 8 $$
Answer: The ladder reaches 8 metres up the wall.
Key Takeaway
Look for triangles in the world around you! Any time you have a diagonal distance and two straight lines (like a TV screen's diagonal, or a path across a park), you can use Pythagoras' Theorem to find missing lengths.
Fun Zone: Pythagorean Triples (Enrichment)
This is a cool little shortcut. A Pythagorean Triple is a special set of three whole numbers that work perfectly in the theorem. There are no decimals involved!
The most famous triple is: (3, 4, 5)
Let's check it: $$3^2 + 4^2 = 9 + 16 = 25$$. And $$5^2 = 25$$. It works!
Here are some other common ones to remember:
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)
You can even make new triples by multiplying a whole triple by the same number!
For example, take (3, 4, 5) and multiply by 2: You get (6, 8, 10). Let's check: $$6^2 + 8^2 = 36 + 64 = 100$$. And $$10^2 = 100$$. It works too!
Did You Know? A Bit of History
The theorem is named after an ancient Greek mathematician called Pythagoras. He and his followers, known as the Pythagoreans, were one of the first groups to write down a formal proof for this relationship around 500 B.C.
However, historians have found evidence that people in ancient Babylon, Egypt, and China knew about this special property of right-angled triangles over 1,000 years before Pythagoras! They used it to design buildings and measure land.