Navigating the Right Triangle: Simplifying Pythagorean Theory

Right triangles are like those trustworthy friends who never let you down—they each have their roles, with the hypotenuse shining as the star of the show. So, let’s get into this little adventure of equations and side lengths while giving you a solid grasp on the Pythagorean theorem, which can seriously help you on your ASVAB practice test.

Okay, picture this: you've got a right triangle with a hypotenuse (that’s the longest side) measuring 14 units, and one of the other sides (let’s call it side1) taking up 9 units. Your mission, should you choose to accept it, is to figure out the length of the remaining side (we'll refer to it as side2, or maybe just x to keep things simple).

Now, before you dive into numbers, let’s remind ourselves of the Pythagorean theorem—this essential gem in math. It's like the rulebook for right triangles and tells us that the square of the hypotenuse equals the sum of the squares of the other two sides. In other words, if we express this mathematically, it looks like this:

hypotenuse² = side1² + side2²

Let’s tidy things up a bit. When we're trying to figure out what side2 (or x) is, we can rearrange our magical equation to find it. Here's how it goes:

x² = hypotenuse² - side1²

Perfect! By plugging in our known values, we’re on our way. For our specific triangle, substitute the numbers, and voilà! It morphs into:

x² = 14² - 9²

Now it’s just a matter of doing the math. Doing the calculations in your mind or on paper, you start with squaring those lengths. We’ve got:

14² = 196 9² = 81,

So, if we subtract those two, we land at:

x² = 196 - 81

Without getting lost in numbers, that simplifies to:

x² = 115

And, to find x, you just need to take the square root of 115. That’s your answer! This is a fantastic way to showcase how the Pythagorean theorem works and how it can help you neatly solve for the unknown side of a triangle—a crucial skill that comes in handy not just for the ASVAB, but for many math problems.

Now, you might be wondering about the other answer choices—it’s totally natural to be curious. When we look at those options you were given:

A. x² = 14² - 9² B. x² = 14² + 9² C. x² = 14 + 9 D. x² = 14 - 9

Choice A is the correct one. It perfectly follows the Pythagorean theorem. On the other hand, B gets it mixed up by adding instead of subtracting, while C and D veer off into mathematical territory that won’t get you closer to x—sorry, folks!

And hey, as you are getting prepared for the ASVAB, remember that math isn't just a series of equations; it’s a way of establishing relationships and understanding your world better. Practice these principles, familiarize yourself with related concepts, and you’ll feel more comfortable tackling those questions when they come up. Remember, it’s all about building confidence and knowledge as you gear up for that test!