Square Side Length: Diagonal Is 6 Cm
Hey guys! Let's dive into a classic geometry problem: figuring out the side length of a square when we know its diagonal. This is a super useful concept, and once you get the hang of it, you'll be solving these problems in no time! So, let's break it down step by step. We'll explore the relationship between a square's diagonal and its sides, use the Pythagorean theorem, and then nail the correct answer. Grab your thinking caps; it's gonna be fun!
Understanding the Square and Its Diagonal
Before we jump into calculations, let's make sure we're all on the same page with what a square and its diagonal are. A square is a quadrilateral, meaning a four-sided polygon, where all sides are of equal length, and all interior angles are right angles (90 degrees). Think of it like a perfectly symmetrical box. Now, the diagonal is a line segment that connects two non-adjacent vertices (corners) of the square. In simpler terms, it's the line you draw from one corner to the opposite corner, cutting the square into two triangles.
The Diagonal Creates Right Triangles
Here's the cool part: when you draw a diagonal in a square, you create two right-angled triangles. Why is this important? Because right triangles open the door to using the Pythagorean theorem, a fundamental concept in geometry. Each of these right triangles has the diagonal as its hypotenuse (the side opposite the right angle), and the two sides of the square as the other two sides (legs) of the triangle. Knowing this relationship is the key to solving our problem.
The Relationship Between Diagonal and Side
The relationship between the diagonal (d) and the side (s) of a square is expressed by the formula: d = s√2. This formula arises directly from applying the Pythagorean theorem. But where does this formula come from? Let's see the next step, so there are no more doubts.
Applying the Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). Mathematically, it's written as: a² + b² = c², where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides.
Using the Theorem in Our Square
In our case, the diagonal of the square is the hypotenuse (c), and the two sides of the square are the legs (a and b). Since all sides of a square are equal, we can say that a = s and b = s (where 's' is the side length of the square). So, we can rewrite the Pythagorean theorem for our square as:
s² + s² = d²
Where 'd' is the length of the diagonal. Now, let's simplify this equation:
2s² = d²
Solving for the Side Length
To find the side length ('s'), we need to isolate it. Divide both sides of the equation by 2:
s² = d² / 2
Now, take the square root of both sides:
s = √(d² / 2)
This can also be written as:
s = d / √2
To rationalize the denominator (get rid of the square root in the bottom), multiply both the numerator and the denominator by √2:
s = (d√2) / 2
This formula tells us how to calculate the side length of a square if we know the length of its diagonal. It's a direct application of the Pythagorean theorem tailored to the specific geometry of a square.
Solving the Problem: Diagonal = 6 cm
Alright, now that we have the formula, let's plug in the given diagonal length (d = 6 cm) and find the side length ('s').
Plugging in the Value
Using the formula s = d / √2, we substitute d = 6 cm:
s = 6 / √2
Rationalizing the Denominator
To rationalize the denominator, we multiply both the numerator and the denominator by √2:
s = (6√2) / 2
Simplifying the Expression
Now, we simplify the expression:
s = 3√2
So, the side length of the square is 3√2 cm.
Analyzing the Options
Now, we need to look at the options provided and see which one matches our calculated side length. The options were:
A) 3 cm B) 4 cm C) 5 cm D) 6 cm
Comparing with the Calculated Value
Our calculated side length is 3√2 cm. This is approximately equal to 3 * 1.414, which is about 4.24 cm. None of the options (3 cm, 4 cm, 5 cm, or 6 cm) exactly match our calculated value. However, let's re-examine our calculations and the relationship between the diagonal and side length.
Correcting the Approach
Oops! It looks like we made a slight misinterpretation in directly matching the simplified radical form to the answer choices. We need to approximate 3√2 to a decimal to compare effectively with the given options.
3√2 ≈ 3 * 1.414 ≈ 4.24 cm
Choosing the Best Answer
Upon closer inspection, and considering we might need to choose the closest answer, it seems there was no right answer in the options. The correct answer is approximately 4.24cm.
Conclusion
So, while none of the provided options were perfectly correct, the side length of a square with a diagonal of 6 cm is approximately 4.24 cm. Remember, the key to solving these problems is understanding the relationship between the diagonal and the side of the square, and correctly applying the Pythagorean theorem. Keep practicing, and you'll become a geometry master in no time! Well done, guys!