Understanding the Pythagorean Theorem: A Simple Explanation and Proof

Mathematics has always fascinated me, and one of the most iconic and useful theorems I came across is the Pythagorean Theorem. This theorem is fundamental in geometry and has a wide range of applications, from architecture to computer graphics. In this blog, I’ll be explaining the Pythagorean Theorem, how to use it, and even going through a simple geometric proof to help you understand it better.

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What is the Pythagorean Theorem?

The Pythagorean Theorem is a formula that works specifically with right-angled triangles. It states that the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:

$$a^2 + b^2 = c^2$$

Where:

• $a$ and $b$ are the lengths of the two legs (the sides forming the right angle),
• $c$ is the length of the hypotenuse.

This simple formula helps us calculate one of the sides of a right-angled triangle if we know the other two sides.

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Let’s Look at an Example

To better understand how this works, let’s take a look at a practical example. Imagine a right-angled triangle where one leg is 3 units long and the other is 4 units. According to the Pythagorean Theorem, we can find the hypotenuse like this:

                                                          

                 

$$c^2 = 3^2 + 4^2 = 9 + 16 = 25$$

Now, we take the square root of both sides:

$$c = \sqrt{25} = 5$$

So, the hypotenuse is 5 units long. Pretty simple, right?

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Applications of the Pythagorean Theorem

The Pythagorean Theorem isn’t just a cool math trick—it has real-world applications too. Some of the ways this theorem is used in everyday life include:

• Construction: Builders often use the Pythagorean Theorem to check if corners are square or to ensure the structure is level.
• Navigation: Pilots and sailors use the theorem to calculate the shortest route or distance between two locations on a map.
• Computer Graphics: In game development and graphic design, the Pythagorean Theorem is used to calculate distances between points, which is crucial for rendering images and animations.

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A Simple Proof of the Pythagorean Theorem

Now that we know how to use the theorem, let’s look at a simple proof. Here’s a geometric proof that I find straightforward:                                               

1. Imagine a large square with a side length of $a + b$. The area of the square is $(a + b)^2$.
2. Inside the large square, place four identical right-angled triangles, each with legs $a$ and $b$, and a hypotenuse of length $c$. These triangles will leave a smaller square in the center with a side length of $c$.
3. The total area of the four triangles is: $$4 \times \frac{1}{2}ab = 2ab$$
4. The area of the small square in the center is $c^2$.
5. Now, the total area of the large square can be expressed as: $$(a + b)^2 = 2ab + c^2$$
6. Expanding the left side: $$a^2 + 2ab + b^2 = 2ab + c^2$$
7. Canceling the $2ab$ from both sides: $$a^2 + b^2 = c^2$$

And that’s a proof of the Pythagorean Theorem! It’s amazing how something so simple can be proven geometrically.

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Practice Problems to Try

Here are a couple of problems you can try to reinforce your understanding of the Pythagorean Theorem:

Problem 1: A ladder leans against a wall, forming a right-angled triangle with the ground. The ladder is 12 feet long, and the distance from the base of the ladder to the wall is 9 feet. How high is the top of the ladder from the ground?

Solution: Using the Pythagorean Theorem:

                                                

Let's assume a(12ft) as the ladder and the distance between base of the ladder and the wall is b(9ft).

$$c^2 = 12^2 - 9^2 = 144 - 81 = 63$$ $$c = \sqrt{63} \approx 7.94 \text{ feet}$$

Problem 2: A right triangle has one side of length 7 and another side of length 24. What is the length of the hypotenuse?

Solution: Using the Pythagorean Theorem:

Relate this with the same diagram that we used for the first problem,

$$c^2 = 7^2 + 24^2 = 49 + 576 = 625$$ $$c = \sqrt{625} = 25$$

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Conclusion

The Pythagorean Theorem is a cornerstone of geometry that not only simplifies the process of solving problems with right-angled triangles but also plays a huge role in many fields. From architecture to technology, its applications are far-reaching and continue to help us solve complex problems.

If you’re interested in delving deeper into geometry or related topics, I highly recommend exploring trigonometry or vector analysis, where the Pythagorean Theorem forms the foundation for many more advanced concepts.

Feel free to share your thoughts or ask questions in the comments below!