The Pythagorean Theorem is an important concept in geometry. It helps us figure out distances, especially in right triangles. This theorem is super useful in many real-life situations. Let's take a look at how it helps us calculate distances.

What is the Pythagorean Theorem?

The Pythagorean Theorem says that in a right triangle, if you square the length of the hypotenuse (the longest side across from the right angle), it is equal to the sum of the squares of the other two sides.

We can write this as:

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

Here,

• $c$ is the length of the hypotenuse
• $a$ and $b$ are the lengths of the other two sides

Using Coordinates for Distance

When we want to find the distance on a coordinate plane, the Pythagorean Theorem becomes very helpful.

Think about two points: Point A $(x_1, y_1)$ and Point B $(x_2, y_2)$. The distance, $d$, between these points can be seen as the hypotenuse of a right triangle. The legs of this triangle are the differences in their x-coordinates and y-coordinates.

How to Find the Distance

To find the distance, we first look at how the coordinates change:

• Change in x: $x_2 - x_1 = a$
• Change in y: $y_2 - y_1 = b$

According to the Pythagorean Theorem, we can express the distance, $d$, as:

$$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$

Then, we can find the distance by taking the square root of both sides:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Real-Life Example

Let’s say you dropped your phone in a park. You know your coordinates are $(3, 4)$ and your friend’s coordinates are $(7, 1)$. You can use the distance formula to find out how far apart you are.

Just plug in the numbers:

$$d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$

This tells you that you are 5 units away from your friend!

Conclusion

To wrap it up, the Pythagorean Theorem is not just a math rule; it's a handy tool for measuring distances in the real world. Knowing how to use it, especially with coordinate points, can help you solve many everyday problems. So, the next time you need to measure a distance, remember that the Pythagorean Theorem can help you out!