Pythogoras theorem

Habari Mwanafunzi! Ready for a Geometry Adventure?

Ever watched a fundi (artisan) building a house and wondered how they make sure the corners are perfectly square? Or have you ever tried to find the shortest way to cross a football pitch instead of walking around the edges? There's a powerful and ancient mathematical secret behind all of this, and its name is the Pythagoras Theorem. Today, you will become a master of this secret!

So, What is this Pythagoras Theorem?

Don't let the big name scare you! It's a very simple but super useful rule that works only for right-angled triangles. Remember, a right-angled triangle is any triangle that has one corner that is a perfect 90-degree angle, like the corner of your exercise book.

Every right-angled triangle has three special sides:

        /|
       / |
      /  |
     /   |
    c    |  a
   (Hypotenuse) |
    /    |
   /     |
  /______|
      b

• The two shorter 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.
• The Hypotenuse (c): This is the longest side. It's always the one opposite the right angle. Think of it as the 'boss' of the triangle's sides!

The Pythagoras Theorem states that if you take the length of the two shorter sides (a and b), square each one, and then add them together, you will get the exact same number as when you square the longest side (c).

Here is the magic formula:

a² + b² = c²

Image Suggestion: A vibrant, colourful diagram showing a right-angled triangle. On each of the three sides, a square is drawn. The square on side 'a' and the square on side 'b' are filled with small, identical blocks. The image shows these blocks being moved to perfectly fill the square on the longest side, 'c', visually demonstrating that the area of a² plus the area of b² equals the area of c².

Let's Bring it Home: Pythagoras in Kenya!

This isn't just some theory in a textbook; it's used all around us every day!

Imagine a farmer, Mama Njeri, in Kinangop who has a rectangular shamba (farm). Her shamba is 30 metres wide and 40 metres long. If she wants to walk from one corner to the corner diagonally opposite, what is the shortest distance she can walk? Her path, the width, and the length form a perfect right-angled triangle! Pythagoras to the rescue!

Image Suggestion: An aerial photograph of a lush green Kenyan farm (shamba) with plots of maize and sukuma wiki. A dotted line is drawn diagonally from one corner to the other. The length and width are labeled, showing how the diagonal path forms the hypotenuse of a right-angled triangle.

Okay, Let's Do The Maths! (Kupiga Hesabu)

Seeing it in action is the best way to learn. Let's solve two common types of problems.

Example 1: Finding the Longest Side (The Hypotenuse)

A carpenter in Gikomba is making a rectangular table frame. One side is 60 cm and the other is 80 cm. He needs to find the length of the diagonal piece of wood to make the frame strong. Here, the diagonal is the hypotenuse (c).

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

Step 2: Put the numbers you know into the formula.
Let a = 60 cm and b = 80 cm.
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'.
To find 'c', we need to find the number that, when multiplied by itself, gives 10000. This is called the square root (√).
c = √10000
c = 100 cm

Answer: The diagonal piece of wood needs to be 100 cm long.

Example 2: Finding a Shorter Side

Imagine you have a 13-foot ladder and you lean it against a wall. The bottom of the ladder is 5 feet away from the base of the wall. How high up the wall does the ladder reach? Here, the ladder is the hypotenuse (c=13), and the distance from the wall is one of the shorter sides (b=5). We need to find the other shorter side (a).

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

Step 2: Rearrange the formula to find 'a'.
We need to get a² by itself. So, we subtract b² from both sides.
a² = c² - b²

Step 3: Put the numbers you know into the new formula.
a² = 13² - 5²

Step 4: Calculate the squares.
a² = (13 * 13) - (5 * 5)
a² = 169 - 25

Step 5: Do the subtraction.
a² = 144

Step 6: Find the square root to get 'a'.
a = √144
a = 12 feet

Answer: The ladder reaches 12 feet up the wall.

Your Turn to be the Expert! (Zoezi)

Grab a pen and your exercise book. Try solving these problems:

• Question 1: A right-angled triangle has shorter sides of 9 cm and 12 cm. What is the length of its hypotenuse?
• Question 2: The hypotenuse of a right-angled triangle is 17 metres long. One of the shorter sides is 8 metres. What is the length of the other shorter side?
• Question 3 (Challenge!): You walk 5 km East from a matatu stage, then turn and walk 12 km North. What is the straight-line distance from the stage to where you are now?

Conclusion: You've Got the Power!

Congratulations! You have just learned one of the most famous and useful theorems in all of mathematics. You now know the secret that helps builders, architects, designers, and even pilots do their jobs accurately.

Remember these key things:

1. The Pythagoras theorem only works for right-angled triangles.
2. The formula is a² + b² = c².
3. 'c' is always the longest side, the hypotenuse.

Keep practicing, and soon you'll be solving these problems without even breaking a sweat. Well done!

Habari Mwanafunzi! Let's Uncover a Mathematical Superpower!

Ever looked at a rectangular field, or a shamba, and wondered what the shortest way to get from one corner to the opposite corner is? Do you walk along the edges, or do you cut straight across? Today, we are going to learn a magical formula from an ancient Greek genius named Pythagoras. This theorem is like a superpower for solving problems involving distances, heights, and shapes. It's used everywhere, from building houses in Nairobi to designing beautiful Kitenge patterns!

Are you ready to unlock this power? Let's begin!

What is a Right-Angled Triangle?

First things first. The Pythagoras theorem only works with a special kind of triangle: the right-angled triangle. This is any triangle that has one perfect corner, like the corner of your exercise book or a door frame. This corner is exactly 90 degrees.

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

• Hypotenuse (c): This is the longest side. It's always the one opposite the right angle. Think of it as the 'sloping' side.
• 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  (Leg 1)
  (Hypotenuse) |
 /     |
/______|
   b (Leg 2)

The Magic Formula: The Pythagorean Theorem

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

Here is the golden formula:

a² + b² = c²

That's it! With this simple formula, we can find the length of a missing side if we know the lengths of the other two. Let's see it in action with some real Kenyan examples.

Example 1: The Shamba Shortcut

Imagine your family's shamba is a perfect rectangle. It is 40 metres long and 30 metres wide. To get to the other side to fetch water, you can either walk along the two sides (40m + 30m = 70m) or you can take a shortcut straight across the diagonal. How long is that shortcut?

The shortcut, the length, and the width of the shamba form a right-angled triangle! The shortcut is our hypotenuse (c).

> **Image Suggestion:** [A vibrant, sunny aerial view of a rectangular green maize shamba in rural Kenya. A dusty path is clearly marked diagonally across the field, connecting two opposite corners. The path, along with two edges of the shamba, should be highlighted to show a right-angled triangle. Label the sides '30m', '40m', and the diagonal path 'Shortcut (c)'.]

Let's do the hesabu (calculation):

1. Identify the sides:
   • a = 30 metres
   • b = 40 metres
   • c = ? (This is the shortcut we want to find)
2. Write down the formula:

   a² + b² = c²

3. Substitute the values:

   30² + 40² = c²

4. Calculate the squares:

   (30 * 30) + (40 * 40) = c²
   900 + 1600 = c²

5. Add them up:

   2500 = c²

6. Find the square root to get 'c':

   c = √2500
   c = 50 metres

Result: The shortcut across the shamba is 50 metres! By using the Pythagoras theorem, we found that you save 20 metres of walking (70m - 50m). Vizuri sana!

Example 2: The Jua Kali Artisan's Ladder

A jua kali artisan needs to fix a window that is 8 metres high on a wall. For the ladder to be safe, its base must be 6 metres away from the wall. How long must the ladder be?

The wall, the ground, and the ladder form another right-angled triangle. The ladder is the hypotenuse!

> **Image Suggestion:** [A side-view of a brightly painted building in a bustling Kenyan town. A skilled jua kali artisan is climbing a wooden ladder that leans against the wall, reaching for a window. The wall, the ground, and the ladder should be highlighted to form a right-angled triangle. Label the height on the wall '8m', the distance on the ground '6m', and the ladder itself 'Ladder (c)'.]

Let's find the length of the ladder ('c'):

1. IDENTIFY SIDES:
   a = 8 m (height of the wall)
   b = 6 m (distance from the wall)
   c = ? (length of the ladder)

2. FORMULA:
   a² + b² = c²

3. SUBSTITUTE:
   8² + 6² = c²

4. CALCULATE:
   64 + 36 = c²
   100 = c²

5. FIND THE SQUARE ROOT:
   c = √100
   c = 10 metres

The artisan needs a ladder that is exactly 10 metres long. See? This is real-life mathematics!

Finding a Shorter Side

What if you know the longest side (hypotenuse) but are missing one of the shorter sides? No problem! We just rearrange the formula.

If a² + b² = c², then:

a² = c² - b²   OR   b² = c² - a²

Let's say a TV screen is advertised as "39 inches". This measurement is the diagonal (the hypotenuse). If the height of the TV is 15 inches, what is its width?

1. IDENTIFY SIDES:
   c = 39 inches (the diagonal)
   a = 15 inches (the height)
   b = ? (the width we need to find)

2. FORMULA:
   b² = c² - a²

3. SUBSTITUTE:
   b² = 39² - 15²

4. CALCULATE:
   b² = 1521 - 225
   b² = 1296

5. FIND THE SQUARE ROOT:
   b = √1296
   b = 36 inches

So, the width of the TV is 36 inches.

Key Takeaway

The Pythagorean Theorem is a powerful tool for any problem involving right-angled triangles. Just remember the magic formula and what each letter stands for.

• It only works for right-angled triangles.
• c is always the longest side, the hypotenuse.
• The formula is a² + b² = c².

Now, go on and practice! Look for right angles around you—in buildings, on playing fields, in furniture—and see how this ancient theorem is still shaping our world today. You've got this!

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

Ever wondered how a fundi (builder) makes sure the corner of a new house is perfectly square? Or how Google Maps calculates the shortest distance between two points, like from your home to the school gate? It’s not magic, it's Mathematics! Today, we are going to learn one of the most famous and useful rules in all of geometry: the Pythagoras Theorem. Get ready, because by the end of this lesson, you'll be solving problems like a pro!

Who was this Pythagoras guy anyway?

Long, long ago (over 2,500 years!), there was a brilliant Greek mathematician named Pythagoras. He and his students were fascinated by numbers and shapes. They discovered a very special relationship in a specific type of triangle, and it was so important that it was named after him. This rule is a cornerstone of geometry and is used in construction, art, design, and even navigation!

The Key Ingredient: The Right-Angled Triangle

The Pythagoras theorem ONLY works with a special triangle called a right-angled triangle. This is any triangle that has one corner that is a perfect 90-degree angle, just like the corner of your exercise book or the corner of a door.

This triangle has three sides with special names:

• The Hypotenuse (c): This is the superstar of the triangle. It's the longest side, and it is always opposite the right angle.
• The Legs (a and b): These are the other two sides that form the right angle. We can call them 'a' and 'b' in any order.

Let's visualize it:

        /|
       / |
      /  |
     /   |
    c    |  a  (Leg 1)
   (Hypo-|
  tenuse)|
  /      |
 /       |
/________|
    b
  (Leg 2)

Here, the angle at the bottom right is the 90° angle.

The Theorem Itself: The Golden Rule

Pythagoras discovered that if you take the length of the two legs (a and b), square each one, and then add them together, you get the exact same number as when you square the length of the hypotenuse (c).

In simple math language, the formula is:

a² + b² = c²

Where:

• a² means 'a' multiplied by 'a' (a * a)
• b² means 'b' multiplied by 'b' (b * b)
• c² means 'c' multiplied by 'c' (c * c)

This simple formula is your key to solving many geometry puzzles!

Kenyan Example: The Boda Boda Shortcut

Imagine a boda boda rider, Juma, needs to get from the market to a customer's house. He can ride 4 kilometres East along the main road, and then turn and ride 3 kilometres North on another road. But, there's a dusty path that cuts straight across the fields. How long is that shortcut?

Image Suggestion: [A vibrant, stylized illustration of a Kenyan town. A boda boda rider is shown at a crossroads (market). One road goes East for 4km, another goes North for 3km. A dotted line shows a diagonal shortcut path across a green field. Label the roads 'a = 4km' and 'b = 3km', and the shortcut 'c = ?'.]

The two roads form a right angle. The shortcut is the hypotenuse! Let's use Pythagoras to help Juma.

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

Step 2: Substitute the values we know.
Let a = 4 km and b = 3 km.
4² + 3² = c²

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

Step 4: Add the numbers together.
25 = c²

Step 5: Find the square root to get 'c'.
To find 'c', we need to find the number that, when multiplied by itself, gives 25. This is called the square root (√).
√25 = c
5 = c

The shortcut is 5 kilometres long! Juma saves (4 + 3) - 5 = 2 kilometres on his trip!

Finding a Shorter Side

What if you know the longest side (hypotenuse) and one of the shorter legs? Easy! We just rearrange the formula a little. If you're looking for side 'a', the formula becomes:

a² = c² - b²

Example: Sizing a TV

You buy a new TV. The box says it's a "40-inch" TV. This measurement is the diagonal length (the hypotenuse!). If the height of the TV is 24 inches, what is its width?

Let c = 40 inches (the diagonal)
Let a = 24 inches (the height)
We need to find b (the width).

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

Step 2: Substitute the values.
b² = 40² - 24²

Step 3: Calculate the squares.
b² = (40 * 40) - (24 * 24)
b² = 1600 - 576

Step 4: Do the subtraction.
b² = 1024

Step 5: Find the square root.
b = √1024
b = 32

The width of the TV is 32 inches!

Handy Shortcuts: Pythagorean Triples

Some numbers work together perfectly in the Pythagoras theorem. These sets are called Pythagorean Triples. Memorizing a few of these can save you a lot of time in an exam!

• 3, 4, 5 (and its multiples, like 6, 8, 10 or 9, 12, 15)
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

Notice in our boda boda example, we had sides 3 and 4. We could have immediately known the hypotenuse was 5 without doing all the calculations!

Putting it all Together: The Fundi's Secret

A smart fundi building a foundation for a house needs to make sure the corners are perfect 90-degree right angles. A small mistake can make the whole house crooked!

She takes her tape measure and makes a mark 3 metres along one wall from the corner. Then she makes a mark 4 metres along the other wall from the corner. Now, she measures the distance directly between those two marks. If that distance is exactly 5 metres, she knows she has a perfect right angle! She is using the 3-4-5 Pythagorean triple to build a strong and straight house.

Image Suggestion: [A close-up illustration of a construction site foundation. A Kenyan fundi (builder, can be a woman) is holding a tape measure. Show the corner of the foundation with markings at 3m on one side, 4m on the other, and the tape measure stretched diagonally showing exactly 5m. The style should be clean and educational.]

Final Word of Encouragement

You've done brilliantly! The Pythagoras theorem might seem like just a formula, but as you've seen, it's a powerful tool for solving real-world problems all around us in Kenya. From the shamba to the city, from building to navigating, right-angled triangles are everywhere. Keep practicing, stay curious, and you'll master this in no time. Kazi nzuri!