A Rectangular Piece Of Paper Has A Width That Is 3 Inches Less Than Its Length. It Is Cut In Half Along The Diagonal To Form Two Congruent Right Triangles, Each With An Area Of 44 Square Inches. Which Statements Are True? Check All That Apply.- The

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Introduction

In mathematics, the relationship between the length and width of a rectangular piece of paper is a fundamental concept that can be used to solve various problems. When a rectangular piece of paper is cut in half along the diagonal, it forms two congruent right triangles. In this article, we will explore the relationship between the length, width, and area of the rectangular piece of paper and determine which statements are true.

The Relationship Between Length and Width

Let's assume that the length of the rectangular piece of paper is L inches and the width is W inches. We are given that the width is 3 inches less than the length, so we can write an equation:

W = L - 3

This equation represents the relationship between the length and width of the rectangular piece of paper.

The Area of the Rectangular Piece of Paper

The area of the rectangular piece of paper is given by the formula:

Area = L × W

Substituting the equation W = L - 3 into the area formula, we get:

Area = L × (L - 3)

The Area of the Right Triangles

When the rectangular piece of paper is cut in half along the diagonal, it forms two congruent right triangles. The area of each right triangle is given as 44 square inches. Since the two right triangles are congruent, the total area of the rectangular piece of paper is twice the area of one right triangle:

Total Area = 2 × 44 = 88 square inches

Solving for the Length and Width

We can use the area formula to solve for the length and width of the rectangular piece of paper. Substituting the total area into the area formula, we get:

88 = L × (L - 3)

Expanding the equation, we get:

88 = L^2 - 3L

Rearranging the equation, we get:

L^2 - 3L - 88 = 0

This is a quadratic equation in L. We can solve for L using the quadratic formula:

L = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -3, and c = -88. Substituting these values into the quadratic formula, we get:

L = (3 ± √((-3)^2 - 4(1)(-88))) / 2(1)

Simplifying the equation, we get:

L = (3 ± √(9 + 352)) / 2

L = (3 ± √361) / 2

L = (3 ± 19) / 2

This gives us two possible values for L:

L = (3 + 19) / 2 = 11

L = (3 - 19) / 2 = -8

Since the length cannot be negative, we discard the solution L = -8. Therefore, the length of the rectangular piece of paper is L = 11 inches.

Finding the Width

Now that we have found the length, we can find the width using the equation W = L - 3:

W = 11 - 3 = 8

Therefore, the width of the rectangular piece of paper is W = 8 inches.

Checking the Statements

Now that we have found the length and width of the rectangular piece of paper, we can check the statements:

  • The length of the rectangular piece of paper is 11 inches.
  • The width of the rectangular piece of paper is 8 inches.
  • The area of the rectangular piece of paper is 88 square inches.
  • The area of each right triangle is 44 square inches.
  • The width is 3 inches less than the length.

All of these statements are true.

Conclusion

In this article, we explored the relationship between the length, width, and area of a rectangular piece of paper. We used the given information to solve for the length and width of the rectangular piece of paper and checked the statements. We found that all of the statements are true. This problem is a great example of how mathematics can be used to solve real-world problems and how the relationship between length, width, and area can be used to solve various problems.

Frequently Asked Questions

  • Q: What is the relationship between the length and width of a rectangular piece of paper? A: The width is 3 inches less than the length.
  • Q: What is the area of the rectangular piece of paper? A: The area of the rectangular piece of paper is 88 square inches.
  • Q: What is the area of each right triangle? A: The area of each right triangle is 44 square inches.
  • Q: How do you find the length and width of a rectangular piece of paper? A: You can use the given information to solve for the length and width using the quadratic formula.

References

  • [1] "Mathematics for Dummies" by Mary Jane Sterling
  • [2] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [3] "Algebra and Trigonometry" by Michael Sullivan

Further Reading

  • "The Pythagorean Theorem" by Michael S. Klamkin
  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Algebra and Trigonometry" by Michael Sullivan

Introduction

In our previous article, we explored the relationship between the length, width, and area of a rectangular piece of paper. We used the given information to solve for the length and width of the rectangular piece of paper and checked the statements. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the relationship between the length and width of a rectangular piece of paper?

A: The width is 3 inches less than the length.

Q: What is the area of the rectangular piece of paper?

A: The area of the rectangular piece of paper is 88 square inches.

Q: What is the area of each right triangle?

A: The area of each right triangle is 44 square inches.

Q: How do you find the length and width of a rectangular piece of paper?

A: You can use the given information to solve for the length and width using the quadratic formula.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is:

Area = Length × Width

Q: What is the formula for the area of a right triangle?

A: The formula for the area of a right triangle is:

Area = (Base × Height) / 2

Q: How do you find the base and height of a right triangle?

A: You can use the given information to solve for the base and height using the Pythagorean theorem.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that describes the relationship between the lengths of the sides of a right triangle. It states that:

a^2 + b^2 = c^2

where a and b are the lengths of the legs of the right triangle, and c is the length of the hypotenuse.

Q: How do you use the Pythagorean theorem to find the base and height of a right triangle?

A: You can use the Pythagorean theorem to find the base and height of a right triangle by substituting the given values into the formula and solving for the unknown values.

Q: What is the relationship between the length, width, and area of a rectangular piece of paper?

A: The length, width, and area of a rectangular piece of paper are related by the formula:

Area = Length × Width

Q: How do you find the length and width of a rectangular piece of paper?

A: You can use the given information to solve for the length and width using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that describes the relationship between the coefficients of a quadratic equation and the roots of the equation. It states that:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do you use the quadratic formula to find the length and width of a rectangular piece of paper?

A: You can use the quadratic formula to find the length and width of a rectangular piece of paper by substituting the given values into the formula and solving for the unknown values.

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the length and width of a rectangular piece of paper. We provided explanations and examples to help clarify the concepts and formulas involved. We hope that this article has been helpful in understanding the relationship between the length, width, and area of a rectangular piece of paper.

Frequently Asked Questions

  • Q: What is the relationship between the length and width of a rectangular piece of paper? A: The width is 3 inches less than the length.
  • Q: What is the area of the rectangular piece of paper? A: The area of the rectangular piece of paper is 88 square inches.
  • Q: What is the area of each right triangle? A: The area of each right triangle is 44 square inches.
  • Q: How do you find the length and width of a rectangular piece of paper? A: You can use the given information to solve for the length and width using the quadratic formula.

References

  • [1] "Mathematics for Dummies" by Mary Jane Sterling
  • [2] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [3] "Algebra and Trigonometry" by Michael Sullivan

Further Reading

  • "The Pythagorean Theorem" by Michael S. Klamkin
  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Algebra and Trigonometry" by Michael Sullivan