The Square Of A Diagonal Of A Square Is 128 Cm Find The Side Of A Square.
Introduction
In the realm of geometry, squares are a fundamental shape that play a crucial role in various mathematical concepts. One of the most interesting properties of a square is its diagonal, which is a line segment connecting two opposite vertices of the square. In this article, we will delve into the world of squares and explore the relationship between the diagonal and the side length of a square. Specifically, we will investigate the problem of finding the side length of a square when the square of its diagonal is given as 128 cm.
Understanding the Problem
The problem states that the square of the diagonal of a square is 128 cm. This means that if we were to square the length of the diagonal, we would get 128 cm. Mathematically, this can be represented as:
diagonal² = 128 cm
Our goal is to find the length of the side of the square, denoted by 's'. To do this, we need to establish a relationship between the diagonal and the side length of the square.
The Relationship Between Diagonal and Side Length
In a square, the diagonal and the side length are related by 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. In the case of a square, the diagonal is the hypotenuse of a right-angled triangle formed by the side length and the other side length.
Using the Pythagorean theorem, we can write:
diagonal² = s² + s²
Simplifying the equation, we get:
diagonal² = 2s²
Now, we can substitute the given value of diagonal² into the equation:
128 cm = 2s²
Solving for the Side Length
To find the side length 's', we need to isolate 's' in the equation. We can do this by dividing both sides of the equation by 2:
s² = 128 cm / 2
s² = 64 cm
Now, we take the square root of both sides of the equation to find the value of 's':
s = √64 cm
s = 8 cm
Therefore, the side length of the square is 8 cm.
Conclusion
In this article, we explored the relationship between the diagonal and the side length of a square. We used the Pythagorean theorem to establish a relationship between the diagonal and the side length, and then solved for the side length using the given value of the square of the diagonal. The result showed that the side length of the square is 8 cm. This problem demonstrates the importance of understanding the properties of squares and how they can be used to solve real-world problems.
Additional Examples
Here are a few additional examples of how the Pythagorean theorem can be used to solve problems involving squares:
- Example 1: Find the side length of a square when the square of its diagonal is 225 cm.
- Example 2: Find the side length of a square when the square of its diagonal is 400 cm.
- Example 3: Find the side length of a square when the square of its diagonal is 900 cm.
Step-by-Step Solutions
Here are the step-by-step solutions to the additional examples:
Example 1: Find the side length of a square when the square of its diagonal is 225 cm.
- Write the equation using the Pythagorean theorem:
diagonal² = s² + s²
- Simplify the equation:
diagonal² = 2s²
- Substitute the given value of diagonal² into the equation:
225 cm = 2s²
- Divide both sides of the equation by 2:
s² = 225 cm / 2
- Simplify the equation:
s² = 112.5 cm
- Take the square root of both sides of the equation to find the value of 's':
s = √112.5 cm
s = 10.58 cm
Therefore, the side length of the square is 10.58 cm.
Example 2: Find the side length of a square when the square of its diagonal is 400 cm.
- Write the equation using the Pythagorean theorem:
diagonal² = s² + s²
- Simplify the equation:
diagonal² = 2s²
- Substitute the given value of diagonal² into the equation:
400 cm = 2s²
- Divide both sides of the equation by 2:
s² = 400 cm / 2
- Simplify the equation:
s² = 200 cm
- Take the square root of both sides of the equation to find the value of 's':
s = √200 cm
s = 14.14 cm
Therefore, the side length of the square is 14.14 cm.
Example 3: Find the side length of a square when the square of its diagonal is 900 cm.
- Write the equation using the Pythagorean theorem:
diagonal² = s² + s²
- Simplify the equation:
diagonal² = 2s²
- Substitute the given value of diagonal² into the equation:
900 cm = 2s²
- Divide both sides of the equation by 2:
s² = 900 cm / 2
- Simplify the equation:
s² = 450 cm
- Take the square root of both sides of the equation to find the value of 's':
s = √450 cm
s = 21.21 cm
Therefore, the side length of the square is 21.21 cm.
Final Thoughts
Introduction
In our previous article, we explored the relationship between the diagonal and the side length of a square. We used the Pythagorean theorem to establish a relationship between the diagonal and the side length, and then solved for the side length using the given value of the square of the diagonal. In this article, we will answer some of the most frequently asked questions about the square of a diagonal of a square.
Q&A
Q: What is the relationship between the diagonal and the side length of a square?
A: The diagonal and the side length of a square are related by 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. In the case of a square, the diagonal is the hypotenuse of a right-angled triangle formed by the side length and the other side length.
Q: How do I find the side length of a square when the square of its diagonal is given?
A: To find the side length of a square when the square of its diagonal is given, you can use the Pythagorean theorem. First, write the equation using the Pythagorean theorem:
diagonal² = s² + s²
Simplify the equation:
diagonal² = 2s²
Substitute the given value of diagonal² into the equation:
diagonal² = 2s²
Divide both sides of the equation by 2:
s² = diagonal² / 2
Take the square root of both sides of the equation to find the value of 's':
s = √(diagonal² / 2)
Q: What if the square of the diagonal is not a perfect square?
A: If the square of the diagonal is not a perfect square, you can still find the side length of the square using the Pythagorean theorem. However, you will need to use a calculator to find the square root of the result.
Q: Can I use the Pythagorean theorem to find the diagonal of a square when the side length is given?
A: Yes, you can use the Pythagorean theorem to find the diagonal of a square when the side length is given. First, write the equation using the Pythagorean theorem:
diagonal² = s² + s²
Simplify the equation:
diagonal² = 2s²
Take the square root of both sides of the equation to find the value of 'diagonal':
diagonal = √(2s²)
Q: What if I have a square with a diagonal of 10 cm and a side length of 5 cm?
A: If you have a square with a diagonal of 10 cm and a side length of 5 cm, you can use the Pythagorean theorem to verify that the diagonal is indeed 10 cm. First, write the equation using the Pythagorean theorem:
diagonal² = s² + s²
Simplify the equation:
diagonal² = 2s²
Substitute the given values into the equation:
10² = 2(5)²
Simplify the equation:
100 = 50
This shows that the diagonal is indeed 10 cm.
Q: Can I use the Pythagorean theorem to find the area of a square when the side length is given?
A: Yes, you can use the Pythagorean theorem to find the area of a square when the side length is given. The area of a square is given by the formula:
Area = s²
where 's' is the side length of the square. You can use the Pythagorean theorem to find the side length of the square, and then use the formula to find the area.
Q: What if I have a square with a side length of 6 cm and a diagonal of 8 cm?
A: If you have a square with a side length of 6 cm and a diagonal of 8 cm, you can use the Pythagorean theorem to verify that the diagonal is indeed 8 cm. First, write the equation using the Pythagorean theorem:
diagonal² = s² + s²
Simplify the equation:
diagonal² = 2s²
Substitute the given values into the equation:
8² = 2(6)²
Simplify the equation:
64 = 72
This shows that the diagonal is not indeed 8 cm. This is because the Pythagorean theorem only works for right-angled triangles, and a square is not a right-angled triangle.
Conclusion
In conclusion, the Pythagorean theorem is a powerful tool for solving problems involving squares. By understanding the relationship between the diagonal and the side length of a square, we can use the theorem to find the side length of a square when the square of its diagonal is given. The Q&A section of this article provides answers to some of the most frequently asked questions about the square of a diagonal of a square.