Mastering Diagonals: Understanding the Square Playground Problem

What is the length of a diagonal of a square playground with a perimeter of 120 yards?

• A. 60√2 yd
• B. 90√2 yd
• C. 45 yd
• D. 30√2 yd

Answer: (D)

To find the length of the diagonal of a square playground with a given perimeter, we first need to determine the side length of the square. The perimeter of a square is calculated using the formula: \[ \text{Perimeter} = 4 \times \text{side length} \] Given the perimeter of the square is 120 yards, we can set up the equation: \[ 120 = 4 \times \text{side length} \] To find the side length, we divide both sides by 4: \[ \text{side length} = \frac{120}{4} = 30 \text{ yards} \] Next, we use the relationship between the side length and the diagonal of a square. The diagonal \(d\) of a square can be calculated using the formula: \[ d = \text{side length} \times \sqrt{2} \] Substituting the side length we found: \[ d = 30 \times \sqrt{2} \] Thus, the length of the diagonal is: \[ d = 30\sqrt{2} \text{ yards} \] This means the correct answer is that the length of the diagonal is 30√2 yards

Understanding the diagonal of a square is more than just a formula — it’s about visualizing the geometry of the world around us. Have you ever strolled through a playground and noticed how the paths create unique angles? This seemingly simple question can actually help you appreciate those angles and how mathematical principles apply to real life.

Let's dig into a common problem seen in college math placement tests: finding the length of the diagonal of a square playground when given its perimeter. Sound intriguing? Don’t worry, I’ll break it down so it clicks.

First Things First: What’s the Perimeter?

The perimeter of any square is like its outer boundary. Picture it as the fence surrounding a playground. The formula to determine the perimeter is pretty straightforward:

[

\text{Perimeter} = 4 \times \text{side length}

]

Here’s the kicker: if the perimeter is 120 yards — like the distance you’d walk around that playground — we can set up our equation:

[

120 = 4 \times \text{side length}

]

Now, let’s solve for the side length. Just grab your calculator (or, you know, rely on that math brain of yours):

[

\text{side length} = \frac{120}{4} = 30 \text{ yards}

]

So, each side of the square playground is 30 yards long. Imagine the space — perfect for all those swings, slides, and maybe even a stickball game!

Time to Find the Diagonal

Next up, we’re diving into the diagonal. The diagonal in a square is that nice straight line cutting from one corner to the opposite corner. But how do we calculate it? Luckily, there’s a formula for that, too:

[

d = \text{side length} \times \sqrt{2}

]

So, we take our side length of 30 yards and plug it into the formula:

[

d = 30 \times \sqrt{2}

]

When we calculate that out, we find:

[

d = 30\sqrt{2} \text{ yards}

]

And guess what? That's our final answer! This means the diagonal length of our charming playground is (30\sqrt{2}) yards.

Why Does This Matter?

You might be wondering, “Why should I care about a playground diagonal?” It's not just about knowing the math; it's about how concepts like these appear in everyday life — from architecture to art to even video game design. Being skilled in these calculations can seriously help demystify the math you encounter in your college courses. Not to mention, understanding geometry sharpens your problem-solving skills and analytical thinking!

As you prepare for your college math placement test, consider this simple question a springboard into the wonderful, geometric world. Practice solving similar problems, and don't hesitate to ask for help when you're stuck—whether from a friend, a teacher, or even a helpful online resource. Every bit of practice brings you closer to mastery. Happy learning!