The Pythagorean Theorem is a super useful tool for figuring out heights in triangles, especially right triangles. Let’s go through it step by step!

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 opposite the right angle), it will equal the sum of the squares of the other two sides.

In simple math terms, we write it like this:

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

Here, $a$ and $b$ are the two shorter sides (legs) of the triangle, and $c$ is the hypotenuse.

How to Use It to Find Height

The Situation

Imagine you have a right triangle. You know the length of one leg and the hypotenuse, and you need to find the length of the other leg, which represents the height.

Here’s How to Do It

1. Know What You Have: Let’s say the hypotenuse $c$ is 10 units long, and one leg $a$ is 6 units long. You want to find the other leg $b$, which is the height.

2. Write the Equation: We can use the Pythagorean Theorem with the numbers we have:

   $$6^2 + b^2 = 10^2$$

3. Solve for $b$:

   • First, calculate $6^2$ and $10^2$:
     • $6^2 = 36$
     • $10^2 = 100$
   • Now plug these values into the equation: $$36 + b^2 = 100$$
   • Next, subtract 36 from both sides: $$b^2 = 100 - 36$$ $$b^2 = 64$$
   • Finally, take the square root of both sides to find $b$: $$b = 8$$

So, you’ve found that the height of the triangle is 8 units!

In Conclusion

Using the Pythagorean Theorem like this is pretty easy and helps you understand how shapes relate to one another in geometry. It’s amazing how one simple formula can help solve all kinds of real-life problems, whether it’s in buildings or engineering!