Pythogoras theorem

Unlocking the Secrets of the Right-Angled Triangle!

Habari mwanafunzi! Ever taken a shortcut across a field instead of walking around the edges? You probably saved some time and energy, right? What you did was use a real-life example of the Pythagoras Theorem, even without knowing it! This powerful idea was discovered by an ancient Greek thinker named Pythagoras, but today, we are going to make it our own and see how it applies right here in Kenya.

This theorem is like a superpower for anyone working with right-angled triangles. So, let's get ready to become geometry superheroes!

First Things First: Meet the Right-Angled Triangle Family

Before we use the magic formula, we need to know the players. The Pythagoras theorem ONLY works for a special kind of triangle: the right-angled triangle. This is any triangle with one angle that is exactly 90 degrees, like the corner of your exercise book or the corner of a room.

The sides of this triangle have special names:

• The Hypotenuse (c): This is the "boss"! It's the longest side and is always found opposite the right angle. It's the shortcut you take across the field.
• The Other Two Sides (a and b): These are the "legs" of the triangle (we can call them miguu!). They are the two sides that form the right angle. It doesn't matter which one you call 'a' and which one you call 'b'.

Let's visualize it:

        /|
       / |
      /  |
     /   |
   c/    |a
   /     |
  /      |
 /_______|
     b

Here:
'c' is the Hypotenuse.
'a' and 'b' are the other two sides.
The small square in the corner shows the 90° angle.

The Magic Formula: a² + b² = c²

Here it is, the heart of the theorem! It looks simple, but it's incredibly powerful. In plain English, it says:

"If you take one of the shorter sides and square it, add it to the square of the other shorter side, you will get the exact same number as the square of the hypotenuse."

Formula: a² + b² = c² 

> **Image Suggestion:** [A colourful, clear diagram of a right-angled triangle. On side 'a' (length 3), a 3x3 grid of squares is drawn. On side 'b' (length 4), a 4x4 grid of squares is drawn. On the hypotenuse 'c' (length 5), a 5x5 grid of squares is drawn. The 9 squares from side 'a' and the 16 squares from side 'b' are shown flying over and perfectly fitting into the 25-square grid on the hypotenuse, visually proving that 9 + 16 = 25.]

Let's Do Some Maths! Finding the Hypotenuse (The Longest Side)

Imagine you are helping a farmer in Kericho to fence a rectangular shamba. The shamba is 30 metres long and 40 metres wide. The farmer wants to put a diagonal fence from one corner to the opposite corner. How long should that fence be?

The sides of the shamba and the diagonal fence form a right-angled triangle!

        /|
       / |
      /  |
  c  /   | 40m (a)
    /    |
   /     |
  /______|
    30m (b)

Let's find 'c' step-by-step:

Step 1: Write down the formula.
a² + b² = c²

Step 2: Substitute the values you know.
Let a = 40, b = 30.
40² + 30² = c²

Step 3: Calculate the squares.
(40 * 40) + (30 * 30) = c²
1600 + 900 = c²

Step 4: Add the numbers together.
2500 = c²

Step 5: Find the square root to get 'c'.
To undo the square (²), we find the square root (√).
√2500 = c
50 = c

Answer: The farmer needs a diagonal fence that is 50 metres long.

Notice how the hypotenuse (50m) is longer than the other two sides (30m and 40m). Your answer must always make sense!

Working Backwards: Finding a Shorter Side

What if you know the longest side and need to find a shorter one? Easy! We just rearrange the formula.

A fundi (craftsman) is using a 10-foot ladder to paint a wall. For safety, he places the bottom of the ladder 6 feet away from the base of the wall. How high up the wall does the ladder reach?

Here, the ladder is the hypotenuse (c = 10 feet), and the distance from the wall is one of the shorter sides (b = 6 feet). We need to find the height on the wall (a).

Wall | /
     |/
     | /
  a  |/  10ft (c)
     | /
     |/
     |_______
       6ft (b)

Let's use a rearranged formula:

Step 1: Start with the original formula.
a² + b² = c²

Step 2: Rearrange it to find 'a²'.
a² = c² - b²

Step 3: Substitute the values.
a² = 10² - 6²

Step 4: Calculate the squares.
a² = (10 * 10) - (6 * 6)
a² = 100 - 36

Step 5: Do the subtraction.
a² = 64

Step 6: Find the square root to get 'a'.
a = √64
a = 8

Answer: The ladder reaches 8 feet high up the wall.

> **Image Suggestion:** [A vibrant, sunny photo of a Kenyan construction site ('jua kali' workshop). A 'fundi' in overalls is holding a large metal square tool against a partially built brick wall to ensure the corner is a perfect 90-degree angle. The scene is active but the focus is on the fundi's precise action.]

Key Takeaways & Your Challenge!

As you can see, the Pythagoras theorem is not just for exams! It's used every day in construction, farming, design, and even navigation. Remember these key points:

• It only works for right-angled triangles.
• The formula is a² + b² = c².
• 'c' is ALWAYS the hypotenuse, the longest side opposite the right angle.
• To find a short side, rearrange the formula to a² = c² - b².
• Don't forget the final step: find the square root!

Now, it's your turn to be the expert!

A flagpole at your school is 12 metres tall. A strong wire is attached from the very top of the pole to a point on the ground that is 5 metres away from the base of the pole. How long is the wire?

Work it out in your book. You've got this! Keep practicing, and you will master this amazing tool. Mbele Pamoja!

Habari Mwanafunzi! Let's Uncover a Mathematical Secret!

Ever seen a fundi (artisan) building a house and wondered how they make sure the corners are perfectly square, at a perfect 90 degrees? Or how a farmer can measure a diagonal path across their shamba (farm) without actually walking it? They use a powerful secret from ancient times, a piece of math magic called the Pythagoras Theorem. Today, you will become a master of this secret!

Don't worry, it's easier than it sounds. By the end of this lesson, you'll be solving problems like a pro. Let's begin!

First Things First: Meet the Right-Angled Triangle

The Pythagoras theorem *only* works with a special kind of triangle: the right-angled triangle. This is any triangle that has one corner which is a perfect square corner, like the corner of your exercise book. We call this a 90-degree angle.

The sides of this triangle have special names:

• The Hypotenuse (c): This is the superstar of the triangle. It's always the longest side, and it's always opposite the right angle.
• The Other Two Sides (a and b): These are the two sides that meet to form the right angle. We can call them the 'legs' of the triangle.

        /|
       / |
      /  |
     /   |
  c /    | a
   /     |
  /      |
 /_______|
     b

Here:
'c' is the Hypotenuse.
'a' and 'b' are the other two sides.
The small square in the corner shows the right angle (90°).

The Magic Formula: a² + b² = c²

A very clever Greek man named Pythagoras discovered something amazing. If you take the length of side 'a' and square it (multiply it by itself), and you do the same for side 'b', the sum of those two numbers will be equal to the square of the hypotenuse, 'c'.

So, the magic formula is:

a² + b² = c²

This means if you draw a square on each side of the triangle, the area of the two smaller squares will add up to the area of the biggest square (the one on the hypotenuse)!

Image Suggestion: A vibrant, colorful diagram showing a right-angled triangle. On side 'a', there's a blue square with area 'a²'. On side 'b', there's a green square with area 'b²'. On the hypotenuse 'c', there's a large yellow square with area 'c²'. Arrows show that the blue and green squares' areas combine to equal the yellow square's area. The style should be simple and educational.

Let's Calculate! Finding the Longest Side (Hypotenuse)

Imagine you need to cross a field. You can walk along the two sides, or take a shortcut straight across. Let's find the length of that shortcut!

Scenario: Maina wants to fence his triangular plot of land in Limuru. One side runs 3 kilometers North, and the adjacent side runs 4 kilometers East. What is the length of the diagonal fence (the hypotenuse) he needs?

Step 1: Identify your 'a', 'b', and 'c'.
a = 3 km
b = 4 km
c = ? (This is the hypotenuse we want to find)

Step 2: Write down the formula.
a² + b² = c²

Step 3: Substitute the values into the formula.
3² + 4² = c²

Step 4: Do the math.
(3 * 3) + (4 * 4) = c²
9 + 16 = c²
25 = c²

Step 5: Find 'c' by finding the square root of 25.
c = √25
c = 5

Answer: Maina needs a diagonal fence that is 5 kilometers long. Sawa?

Finding a Shorter Side

What if you already know the length of the longest side (the hypotenuse) but need to find one of the shorter sides? We just need to rearrange our magic formula a little bit!

Scenario: A KPLC (Kenya Power) worker leans a 10-meter ladder against a wall. The base of the ladder is 6 meters away from the bottom of the wall. How high up the wall does the ladder reach?

        /|
       / |
10 m  /  | ? (a)
(c)  /   |
    /    |
   /_____|
     6 m
     (b)

Here, the ladder is the hypotenuse (c), the distance from the wall is one side (b), and the height on the wall is the other side (a).

Step 1: Identify your 'a', 'b', and 'c'.
a = ? (The height we want to find)
b = 6 m
c = 10 m

Step 2: Write down the formula and rearrange it to find 'a'.
Original: a² + b² = c²
Rearranged: a² = c² - b²  (We just moved b² to the other side)

Step 3: Substitute the values.
a² = 10² - 6²

Step 4: Do the math.
a² = (10 * 10) - (6 * 6)
a² = 100 - 36
a² = 64

Step 5: Find 'a' by finding the square root of 64.
a = √64
a = 8

Answer: The ladder reaches 8 meters high up the wall. See? You can do it!

Image Suggestion: A realistic illustration of a Kenyan construction worker (fundi) on a site in Nairobi. He is using a large carpenter's square to check the corner of a foundation. A ghostly right-angled triangle is overlaid on the corner, with the sides labeled 'a', 'b', and 'c' to show the practical application of the Pythagoras theorem.

Why is this important in Kenya? (And everywhere else!)

This isn't just for your math exam! People use Pythagoras's theorem every day:

• Construction: To make sure walls, windows, and foundations are perfectly square.
• Navigation: A pilot or a matatu driver can calculate the shortest distance between two points.
• Farming: To calculate the size of a piece of land or the length of a fence.
• Art and Design: To get the right proportions in drawings and designs.

Mazoezi (Practice Time!)

Try these on your own. Remember the formula and the steps!

1. A boda boda rider travels 8 km south, then turns and travels 6 km west. What is the shortest distance, in a straight line, back to his starting point?
2. A flagpole is 15 meters tall. A support rope is attached from the top of the pole to a point on the ground 8 meters from the base. How long is the support rope?

Well done for making it this far! The Pythagoras theorem is a fundamental tool in mathematics. Practice it, and you will find it incredibly useful. You are now officially a Pythagoras hero! Keep up the great work.

Habari Mwanafunzi! Let's Unlock the Magic of Triangles!

Have you ever tried to take a shortcut across a field instead of walking along the paths? Or have you seen a fundi (artisan) lean a ladder against a wall? You were actually seeing a real-life maths problem in action! Today, we are going to learn a powerful secret from an ancient Greek maths guru named Pythagoras. This secret is a special rule for right-angled triangles, and it's so useful, you'll see it everywhere from construction sites in Nairobi to farming in the Rift Valley.

By the end of this lesson, you will be able to:

• Identify the sides of a right-angled triangle.
• State and use the Pythagoras theorem to find unknown sides.
• Apply this knowledge to solve real-world problems.

Meet the Star of Our Show: The Right-Angled Triangle

First things first, let's get to know our main character. A right-angled triangle is any triangle that has one perfect corner, like the corner of your exercise book or a window frame. This special corner is exactly 90 degrees (90°).

This triangle has three special sides with fancy names:

• The two sides that form the right angle (the "L" shape) are called the shorter sides. We can label them a and b.
• The longest side, which is always opposite the right angle, is the boss! It's called the hypotenuse, and we label it c.

        /|
       / |
      /  |
   c /   | a
    /    |
   /_____|
      b

c = Hypotenuse (The longest side)
a & b = The other two shorter sides

Image Suggestion: A vibrant, colourful graphic showing a right-angled triangle. The right-angle corner is highlighted with a square symbol. The hypotenuse 'c' is a bright, bold colour, while sides 'a' and 'b' are other distinct colours. Animated arrows point from the right-angle to the hypotenuse, labeling it "The Boss!".

The Magic Formula: The Pythagoras Theorem

Pythagoras discovered a fantastic relationship between these three sides. It's a formula that works every single time for any right-angled triangle. Are you ready for it?

The theorem states: "The square of side 'a' plus the square of side 'b' is equal to the square of the hypotenuse 'c'."

In maths language, it looks like this:

a² + b² = c²

This simple formula is the key to solving so many problems! Let's see how to use it.

Finding the Longest Side (The Hypotenuse)

Imagine you are walking from the GPO in Nairobi. You walk 3 kilometres East, then turn North and walk 4 kilometres. What is the shortest straight-line distance back to where you started? You've just made a right-angled triangle with your path!

              FINISH
                /\
                |
                | 4 km (a)
                |
 START O--------|
          3 km (b)

Shortest distance (c) = ?

Let's use our magic formula to find 'c'.

Step 1: Write down the formula.
a² + b² = c²

Step 2: Put in the numbers you know.
(a = 4 km, b = 3 km)
4² + 3² = c²

Step 3: Calculate the squares.
(4² = 4x4 = 16)
(3² = 3x3 = 9)
16 + 9 = c²

Step 4: Add the numbers together.
25 = c²

Step 5: Find the square root to get 'c'.
To find 'c', we do the opposite of squaring, which is finding the square root (√).
c = √25
c = 5

So, the shortest distance back is 5 kilometres!

Working Backwards: Finding a Shorter Side

What if you already know the longest side? For instance, a fundi has a ladder that is 5 metres long. He leans it against a wall. The base of the ladder is 3 metres away from the wall. How high up the wall does the ladder reach?

Image Suggestion: A simple, clear diagram of a ladder leaning against a brick wall, forming a right-angled triangle. The ladder is labeled 'c = 5m', the distance from the wall is 'b = 3m', and the height on the wall is 'a = ?'. The style should be like a textbook illustration.

Here, the ladder is the hypotenuse (c=5m), and the ground is one of the shorter sides (b=3m). We need to find the other shorter side, the height on the wall (a).

We just need to rearrange our formula a little bit:

If: a² + b² = c²
Then: a² = c² - b²

Now let's solve it!

Step 1: Write down the rearranged formula.
a² = c² - b²

Step 2: Put in the numbers you know.
(c = 5 m, b = 3 m)
a² = 5² - 3²

Step 3: Calculate the squares.
(5² = 5x5 = 25)
(3² = 3x3 = 9)
a² = 25 - 9

Step 4: Subtract the numbers.
a² = 16

Step 5: Find the square root to get 'a'.
a = √16
a = 4

The ladder reaches 4 metres high up the wall!

Pythagorean Triples: The Maths "Cheat Codes"

Some numbers are special friends and always work together perfectly in the Pythagoras theorem, giving you nice, whole number answers. We call these Pythagorean Triples. If you can remember them, you can solve some problems much faster!

Here are the most common ones:

• 3, 4, 5 (Like our Nairobi GPO example!)
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

Even their multiples work! For example, if you double the 3, 4, 5 triple, you get 6, 8, 10. Let's check: 6² + 8² = 36 + 64 = 100, and √100 = 10. It works!

So, Why Is This Important?

This isn't just classroom maths; it's life maths! Pythagoras is used all the time.

A farmer in a shamba (farm) has a rectangular plot of land. She wants to build a fence diagonally from one corner to the opposite corner. If she knows the length and width of her plot, she can use the Pythagoras theorem to calculate the exact length of the diagonal fence she needs to build. This saves her time and money because she buys exactly the right amount of wire!

It's also used in:

• Construction: To make sure walls are perfectly square (90 degrees).
• Navigation: To find the shortest distance between two points (like a pilot or a ship captain).
• Design: When creating anything from furniture to video game graphics.

Your Turn to Be the Guru!

You have now learned the powerful secret of Pythagoras! You can find a missing side of any right-angled triangle. You have a new superpower.

Challenge: The screen of your smartphone has a diagonal of 15 cm and a width of 9 cm. Can you use the Pythagoras theorem to find its height? Give it a try!

Keep practising, and you'll become a master in no time. Kazi nzuri!