A Square Tile Is Twelve Inches On A Side. It Is Cut Across The Diagonal To Form Two Right Triangles. What Is The Length Of The Diagonal Cut?

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Introduction

When a square tile is cut across the diagonal, it forms two right triangles. The length of the diagonal cut is a crucial aspect of geometry, and understanding it is essential for various mathematical applications. In this article, we will delve into the concept of the diagonal cut and explore the methods to calculate its length.

The Concept of the Diagonal Cut

A square tile with sides of 12 inches is a classic example of a geometric shape. When it is cut across the diagonal, it forms two right triangles. The diagonal cut is the line that divides the square into two equal parts, creating two right angles. This concept is fundamental in geometry and is used in various mathematical applications, including trigonometry and calculus.

Calculating the Length of the Diagonal Cut

To calculate the length of the diagonal cut, we can use 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.

Mathematically, this can be expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

In the case of the square tile cut across the diagonal, the two sides are equal in length, and the diagonal cut is the hypotenuse. Therefore, we can write:

a^2 + a^2 = c^2

Simplifying the equation, we get:

2a^2 = c^2

Since the length of the side of the square tile is 12 inches, we can substitute this value into the equation:

2(12)^2 = c^2

Expanding the equation, we get:

288 = c^2

Taking the square root of both sides, we get:

c = √288

Simplifying the expression, we get:

c = √(144 × 2)

c = √144 × √2

c = 12√2

Therefore, the length of the diagonal cut is 12√2 inches.

Understanding the Concept of √2

The concept of √2 is a fundamental aspect of mathematics. It is an irrational number that represents the square root of 2. In the context of the diagonal cut, √2 is used to calculate the length of the hypotenuse.

The value of √2 is approximately 1.4142. This value is used in various mathematical applications, including trigonometry and calculus.

Conclusion

In conclusion, the length of the diagonal cut of a square tile can be calculated using the Pythagorean theorem. By substituting the length of the side of the square tile into the equation, we can determine the length of the diagonal cut. The concept of √2 is a fundamental aspect of mathematics and is used in various mathematical applications.

Frequently Asked Questions

Q: What is the length of the diagonal cut of a square tile with sides of 12 inches?

A: The length of the diagonal cut is 12√2 inches.

Q: How is the length of the diagonal cut calculated?

A: The length of the diagonal cut is calculated using the Pythagorean theorem.

Q: What is the value of √2?

A: The value of √2 is approximately 1.4142.

References

Further Reading

  • [1] "Geometry: A Comprehensive Introduction." By Dan Pedoe. Dover Publications, 2013.
  • [2] "Calculus: Early Transcendentals." By James Stewart. Cengage Learning, 2015.

Glossary

  • Hypotenuse: The side of a right-angled triangle opposite the right angle.
  • Pythagorean Theorem: A mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle.
  • √2: The square root of 2, an irrational number approximately equal to 1.4142.

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Introduction

When a square tile is cut across the diagonal, it forms two right triangles. The length of the diagonal cut is a crucial aspect of geometry, and understanding it is essential for various mathematical applications. In this article, we will delve into the concept of the diagonal cut and explore the methods to calculate its length.

The Concept of the Diagonal Cut

A square tile with sides of 12 inches is a classic example of a geometric shape. When it is cut across the diagonal, it forms two right triangles. The diagonal cut is the line that divides the square into two equal parts, creating two right angles. This concept is fundamental in geometry and is used in various mathematical applications, including trigonometry and calculus.

Calculating the Length of the Diagonal Cut

To calculate the length of the diagonal cut, we can use 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.

Mathematically, this can be expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

In the case of the square tile cut across the diagonal, the two sides are equal in length, and the diagonal cut is the hypotenuse. Therefore, we can write:

a^2 + a^2 = c^2

Simplifying the equation, we get:

2a^2 = c^2

Since the length of the side of the square tile is 12 inches, we can substitute this value into the equation:

2(12)^2 = c^2

Expanding the equation, we get:

288 = c^2

Taking the square root of both sides, we get:

c = √288

Simplifying the expression, we get:

c = √(144 × 2)

c = √144 × √2

c = 12√2

Therefore, the length of the diagonal cut is 12√2 inches.

Understanding the Concept of √2

The concept of √2 is a fundamental aspect of mathematics. It is an irrational number that represents the square root of 2. In the context of the diagonal cut, √2 is used to calculate the length of the hypotenuse.

The value of √2 is approximately 1.4142. This value is used in various mathematical applications, including trigonometry and calculus.

Conclusion

In conclusion, the length of the diagonal cut of a square tile can be calculated using the Pythagorean theorem. By substituting the length of the side of the square tile into the equation, we can determine the length of the diagonal cut. The concept of √2 is a fundamental aspect of mathematics and is used in various mathematical applications.

Frequently Asked Questions

Q: What is the length of the diagonal cut of a square tile with sides of 12 inches?

A: The length of the diagonal cut is 12√2 inches.

Q: How is the length of the diagonal cut calculated?

A: The length of the diagonal cut is calculated using the Pythagorean theorem.

Q: What is the value of √2?

A: The value of √2 is approximately 1.4142.

Q: What is the relationship between the length of the diagonal cut and the length of the side of the square tile?

A: The length of the diagonal cut is equal to the length of the side of the square tile multiplied by √2.

Q: Can the length of the diagonal cut be calculated using other methods?

A: Yes, the length of the diagonal cut can be calculated using other methods, such as the use of trigonometry or the properties of right-angled triangles.

Q: What is the significance of the diagonal cut in geometry?

A: The diagonal cut is a fundamental concept in geometry and is used in various mathematical applications, including trigonometry and calculus.

Q&A Session

Q: I have a square tile with sides of 15 inches. How can I calculate the length of the diagonal cut?

A: To calculate the length of the diagonal cut, you can use the Pythagorean theorem. The formula is:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

In this case, the two sides are equal in length, and the diagonal cut is the hypotenuse. Therefore, we can write:

a^2 + a^2 = c^2

Simplifying the equation, we get:

2a^2 = c^2

Since the length of the side of the square tile is 15 inches, we can substitute this value into the equation:

2(15)^2 = c^2

Expanding the equation, we get:

450 = c^2

Taking the square root of both sides, we get:

c = √450

Simplifying the expression, we get:

c = √(225 × 2)

c = √225 × √2

c = 15√2

Therefore, the length of the diagonal cut is 15√2 inches.

Q: I have a square tile with sides of 20 inches. How can I calculate the length of the diagonal cut?

A: To calculate the length of the diagonal cut, you can use the Pythagorean theorem. The formula is:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

In this case, the two sides are equal in length, and the diagonal cut is the hypotenuse. Therefore, we can write:

a^2 + a^2 = c^2

Simplifying the equation, we get:

2a^2 = c^2

Since the length of the side of the square tile is 20 inches, we can substitute this value into the equation:

2(20)^2 = c^2

Expanding the equation, we get:

800 = c^2

Taking the square root of both sides, we get:

c = √800

Simplifying the expression, we get:

c = √(400 × 2)

c = √400 × √2

c = 20√2

Therefore, the length of the diagonal cut is 20√2 inches.

Conclusion

In conclusion, the length of the diagonal cut of a square tile can be calculated using the Pythagorean theorem. By substituting the length of the side of the square tile into the equation, we can determine the length of the diagonal cut. The concept of √2 is a fundamental aspect of mathematics and is used in various mathematical applications.

References

Further Reading

  • [1] "Geometry: A Comprehensive Introduction." By Dan Pedoe. Dover Publications, 2013.
  • [2] "Calculus: Early Transcendentals." By James Stewart. Cengage Learning, 2015.

Glossary

  • Hypotenuse: The side of a right-angled triangle opposite the right angle.
  • Pythagorean Theorem: A mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle.
  • √2: The square root of 2, an irrational number approximately equal to 1.4142.