Pythogoras theorem

Habari Mwanafunzi! Let's Uncover the Magic of Pythagoras!

Have you ever tried to take a shortcut by cutting across a field instead of walking around the edges? You probably knew it was a shorter distance, but did you ever wonder exactly how much shorter it was? Well, today, we are going to learn a magical piece of mathematics from ancient Greece that helps us answer exactly that kind of question. Get ready to meet the amazing Pythagoras Theorem!

Image Suggestion: A vibrant, colourful illustration of a student in a Kenyan school uniform standing at the corner of a rectangular shamba (farm). They are looking thoughtfully at the diagonal path across the field versus the path along the edges. The paths form a bright, glowing right-angled triangle. The style should be encouraging and slightly cartoonish.

First Things First: The Right-Angled Triangle

The Pythagoras theorem has a favourite shape: the right-angled triangle. This is any triangle that has one perfect corner, like the corner of your exercise book or a window. This special corner is called a right angle, and it measures 90 degrees (90°).

Every right-angled triangle has three sides with special names:

• The Hypotenuse: This is the "boss" of the sides! It's always the longest side and is always found opposite the right angle. We often call it 'c'.
• The Other Two Sides: These are the two sides that form the right angle. We can call them 'a' and 'b'. They are sometimes called the "legs" of the triangle.

Let's look at one right now:

      /|
     / |
    /  |
 c /   | b
  /    |
 /     |
/______|
   a

(The little square in the corner shows the right angle, 90°)

The Magic Formula: a² + b² = c²

A very clever man named Pythagoras discovered a fantastic relationship between these three sides. He found that if you take the two shorter sides ('a' and 'b'), square each one, and then add them together, you get the exact same number as when you square the longest side ('c')!

This gives us the famous theorem:

a² + b² = c²

Where:

• a is the length of one short side.
• b is the length of the other short side.
• c is the length of the hypotenuse.

Example 1: Fencing a Shamba

Let's put this into practice. Imagine you have a small triangular shamba in your compound, and you need to put a fence around it. The two sides that meet at the right angle are 3 metres long and 4 metres long. What is the length of the third, longest side (the hypotenuse)?

Scenario: A farmer has a small triangular plot of land. Side 'a' is 3m and side 'b' is 4m. He needs to find the length of the diagonal side 'c' to buy the right amount of fencing wire.

Let's use our formula!

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

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

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

Step 4: Add the numbers together.
25 = c²

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

Answer: The length of the hypotenuse is 5 metres! You need 5 metres of wire for that side. See? Not so hard!

Example 2: A Ladder Against a Nyumba

What if we already know the longest side? Don't worry, we can still use the theorem. We just need to rearrange it a little.

Scenario: A fundi (artisan) leans a 10-metre long ladder against the wall of a house. The base of the ladder is 6 metres away from the wall. How high up the wall does the ladder reach?

Here, the ladder is the hypotenuse (c = 10m), and the distance from the wall is one of the shorter sides (let's say a = 6m). We need to find the other short side, 'b', which is the height on the wall.

Image Suggestion: A realistic digital painting of a colourful Kenyan house (nyumba) with a wooden ladder leaning against it. The ground, the wall, and the ladder form a clear right-angled triangle. Label the ladder 'c = 10m', the ground distance 'a = 6m', and the height on the wall 'b = ?'.

Our formula is a² + b² = c². To find a shorter side, we subtract!

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

Step 2: Put in the numbers we know.
(c = 10, a = 6)
b² = 10² - 6²

Step 3: Calculate the squares.
(10² = 10 x 10 = 100)
(6² = 6 x 6 = 36)
b² = 100 - 36

Step 4: Subtract the numbers.
b² = 64

Step 5: Find the square root to get 'b'.
b = √64
b = 8

Answer: The ladder reaches 8 metres up the wall!

Tukumbuke (Let's Remember): Key Takeaways

Great job for making it this far! Here are the most important things to remember about the Pythagoras Theorem:

• 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).
• When you are looking for the hypotenuse, you ADD the squares.
• When you are looking for a shorter side, you SUBTRACT the other square from the hypotenuse's square.

You have just learned a very powerful tool in mathematics. It's used in building, art, design, and even in finding your way around. Keep practicing, and soon you'll be a Pythagoras master! Hongera!

Habari Mwanafunzi! Pythagoras Theorem: The Secret of the Right-Angled Triangle!

Welcome to the world of Geometry! Today, we are going to unlock one of the most famous and useful rules in all of mathematics. Ever wondered how a construction worker makes sure a corner is perfectly square? Or how to find the shortest path across a rectangular shamba without a measuring tape for the long distance? The secret lies with a clever Greek fellow named Pythagoras!

This lesson will make you a master of the right-angled triangle. Uko tayari? Let's begin!

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

The Pythagoras theorem ONLY works for a special kind of triangle: the right-angled triangle. In Kenya, you might hear it called a pembetatu mraba. This triangle is special because one of its corners is a perfect 90-degree angle, just like the corner of your exercise book or a window frame.

A right-angled triangle has three sides with special names:

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

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

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

The Magic Formula: The Pythagoras Theorem

Pythagoras discovered a fantastic relationship between these three sides. He found that if you take the length of side 'a' and multiply it by itself (which we call 'squaring'), and you do the same for side 'b', their sum will be equal to the square of the hypotenuse, 'c'.

In simple mathematics, the formula is:

a² + b² = c²

This simple formula is the key to solving so many real-world problems!

Image Suggestion: A vibrant, colourful diagram showing a right-angled triangle. On side 'a', there is a square made of 3x3 smaller squares (total 9). On side 'b', there is a square of 4x4 smaller squares (total 16). On the hypotenuse 'c', there is a larger square of 5x5 smaller squares (total 25), visually demonstrating that 9 + 16 = 25.

Example 1: Kamau's Shamba Shortcut

Kamau has a rectangular shamba (farm). It is 80 metres long and 60 metres wide. Every day, he walks along the edges to get from one corner to the opposite corner. He wants to know the distance if he walks straight across the middle (the diagonal). Can we help him find the shortcut distance?

Yes! The diagonal of the shamba creates a right-angled triangle with the two sides.

• Side 'a' = 60 metres
• Side 'b' = 80 metres
• The shortcut (hypotenuse 'c') = ?

Let's use our formula!

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

Step 2: Substitute the values we know
60² + 80² = c²

Step 3: Calculate the squares
(60 * 60) + (80 * 80) = c²
3600 + 6400 = c²

Step 4: Add the numbers
10000 = c²

Step 5: Find the square root to get 'c'
c = √10000
c = 100 metres

Answer: The shortcut across Kamau's shamba is 100 metres! He saves a lot of walking!

Example 2: The Ladder Against the Wall

Sometimes, we know the longest side (the hypotenuse) but need to find one of the shorter sides.

Amina has a 5-metre ladder. She leans it against a wall. The base of the ladder is 3 metres away from the wall on the ground. How high up the wall does the ladder reach?

Here, the ladder is the hypotenuse! The wall and the ground form the right angle.

• Ladder (hypotenuse 'c') = 5 metres
• Distance from wall (side 'b') = 3 metres
• Height on the wall (side 'a') = ?

We need to rearrange our formula to find 'a'.

If a² + b² = c², then to find a², we subtract b² from both sides:
a² = c² - b²

Now, let's solve Amina's problem.

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

Step 2: Substitute the values
a² = 5² - 3²

Step 3: Calculate the squares
a² = (5 * 5) - (3 * 3)
a² = 25 - 9

Step 4: Do the subtraction
a² = 16

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

Answer: The ladder reaches 4 metres high up the wall.

Image Suggestion: A simple, clear illustration of a Kenyan scene: a wooden ladder leaning against a mabati (corrugated iron) wall. The ground, wall, and ladder should be clearly labeled with '3m', '4m', and '5m' to match the example.

How to Check for a Right Angle (The Converse)

You can also use the theorem to check if a triangle is a right-angled triangle. If a triangle's sides fit the formula `a² + b² = c²`, then it MUST be a right-angled triangle! This is what fundis (craftsmen) use to make sure walls are perfectly straight.

For example, does a triangle with sides 5 cm, 12 cm, and 13 cm have a right angle?

Let's test it. The longest side, 13 cm, must be 'c'.

Check if:  a²   +   b²  =   c²
           5²   +  12²  =  13²
          25   +  144  =  169
               169    =  169

Yes, it works! So, that triangle is a right-angled triangle.

Mazoezi Time! (Your Turn)

Now it's your turn to be the expert. Try solving these problems.

1. A football pitch is 100 metres long and 75 metres wide. What is the length of the diagonal from one corner flag to the other?
2. A boat's sail is shaped like a right-angled triangle. The longest side (hypotenuse) is 17 metres, and the bottom edge is 8 metres. What is the height of the sail?

Kumbuka! (Key Summary)

• The Pythagoras theorem works only for right-angled triangles.
• The formula is a² + b² = c².
• The hypotenuse (c) is always the longest side, opposite the right angle.
• To find a shorter side, you rearrange the formula to a² = c² - b².
• Always remember to find the square root at the end to get the final length!

Hongera! You have just mastered a powerful mathematical tool. Keep practicing, and you will see these triangles everywhere you look. From the roof of a house to the support cables on a radio mast, Pythagoras is all around us in Kenya!

Habari Mwanafunzi! Unlocking the Secrets of the Right-Angled Triangle

Welcome to the exciting world of Geometry! Today, we are going on an adventure to uncover one of the most famous and useful rules in all of mathematics. Imagine you are walking from your home to the *duka* (shop). You can walk along the road, take a turn, and then walk down another road. But what if there is a field, a *shamba*, in between? The shortest path is to cut straight across! But how long is that path? An ancient Greek genius named Pythagoras figured out exactly how to solve this, and today, you will too!

Image Suggestion: A vibrant, cartoon-style illustration of a Kenyan student in school uniform standing at the corner of a green field. A path goes along the two sides of the field to a shop, and another path cuts diagonally across the field. The student is looking thoughtfully at the diagonal path with a question mark above their head.

First Things First: Meet the Right-Angled Triangle

The Pythagoras theorem ONLY works for a special kind of triangle: the right-angled triangle. This is any triangle that has a perfect corner, like the corner of your exercise book or the corner of a room. This corner is exactly 90 degrees (90°).

This special triangle has three sides with special names:

• The Hypotenuse: This is the SUPERSTAR of the triangle. It is the longest side, and you can always find it directly opposite the right angle.
• The Other Two Sides: These are the two shorter sides that meet to form the right angle. We can call them 'a' and 'b'.

Let's draw one:

      /|
     / |
    /  |
   /   |
c /    | a
 /     |
/______|
   b

c = Hypotenuse (the longest, sloping side)
a = One of the shorter sides
b = The other shorter side

The Magic Formula: a² + b² = c²

Pythagoras discovered a magical relationship between these three sides. He found that if you take the lengths of the two shorter sides (a and b), square each one, and then add them together, the answer is exactly the same as the square of the hypotenuse (c)!

In simple language: The square of side 'a' + the square of side 'b' = the square of the hypotenuse 'c'.

Here is the formula you must remember. It's your key to unlocking the secret!

a² + b² = c²

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

Time to get our hands dirty with some numbers. Let's solve a real problem.

Scenario: A farmer has a small triangular shamba. The side going north is 3 metres long, and the side going east is 4 metres long. He wants to put a fence on the diagonal side. How long will the fence be?

     /|
    / |
   /  |
c /   | 3m (a)
 /    |
/_____|
  4m (b)

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

2. Substitute the values we know (a=3, b=4):
   3² + 4² = c²

3. Calculate the squares:
   (3 * 3) + (4 * 4) = c²
   9 + 16 = c²

4. Add the numbers together:
   25 = c²

5. Now, to find 'c' (not c²), we must find the square root of 25:
   c = √25
   c = 5

The farmer needs a fence that is 5 metres long!

See? Not so hard, right? *Hongera!* (Congratulations!) You've just used the Pythagoras theorem.

What If We Know the Hypotenuse? Finding a Shorter Side

Sometimes, the problem is different. You might know the length of the longest side (the hypotenuse) and one of the shorter sides. Don't panic! We can rearrange our magic formula.

To find a short side (let's say 'a'), the formula becomes:

a² = c² - b²

Scenario: A *fundi* (artisan) leans a 10-metre ladder 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?

Image Suggestion: A realistic photo of a colourful building in a Kenyan city like Nairobi or Mombasa. A wooden ladder is leaning against it, forming a right-angled triangle with the wall and the ground. The distances (10m on the ladder, 6m on the ground) are subtly labeled with chalk-like text.

      | /
      |/
      | /  10m (c - the ladder)
 h (a)|/
      | /
      |/
      +-------
        6m (b)

Here, the ladder is the hypotenuse (c=10), the ground is one short side (b=6), and the height on the wall is the other short side we need to find (a=h).

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

2. Substitute the values:
   a² = 10² - 6²

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

4. Do the subtraction:
   a² = 64

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

The ladder reaches 8 metres high up the wall!

Why Does This Matter in Kenya?

You might be wondering, "When will I ever use this?" The answer is: ALL THE TIME! People use it without even knowing the name.

A *mjengo* (construction) worker making sure the corners of a foundation are perfectly square. A pilot calculating the shortest flight path. Even when you're playing football on the field and you kick the ball from one corner to the opposite corner, you've just created a hypotenuse! The Pythagoras theorem helps us understand the world of distances and shapes around us, from the design of the KICC building to the routes a *matatu* takes.

Your Turn to Be the Expert!

You have done so well. Now it is your chance to solve a problem on your own. Read it carefully, draw a small picture to help you, and use the magic formula.

Challenge: A flagpole at your school is 12 metres tall. A strong rope is tied 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 rope?

Take your time, follow the steps, and you will find the answer. You are now a geometry detective, able to find missing distances and solve real-world puzzles. *Kazi nzuri!* (Good work!)