Solving For Dimensions Of A Cut Rectangular Paper A Geometry Problem

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This article explores a fascinating geometric problem involving a rectangular piece of paper cut along its diagonal. The problem presents a scenario where the width of the rectangle is 3 inches less than its length, and the diagonal cut creates two congruent right triangles, each with an area of 44 square inches. Our objective is to determine the true statements that accurately describe the dimensions and properties of this rectangle and the resulting triangles. This involves applying geometric principles, algebraic equations, and problem-solving strategies to analyze the given information and arrive at logical conclusions. Let’s delve into the intricacies of this problem and unravel the dimensions of the cut rectangular paper.

We are given a rectangular piece of paper with a specific relationship between its length and width: the width is 3 inches shorter than the length. This paper is then cut in half along a diagonal line, resulting in two identical right-angled triangles. Each of these triangles has an area of 44 square inches. The core of the problem lies in identifying the correct statements that accurately describe the dimensions of the original rectangle and the properties of the resulting triangles. This requires a blend of geometric understanding, algebraic manipulation, and logical deduction.

To begin solving this problem, let's define our variables. Let 'l' represent the length of the rectangle and 'w' represent its width. According to the problem statement, the width is 3 inches less than the length, so we can write this relationship as an equation:

w = l - 3

Next, we know that the area of each right triangle formed by cutting the rectangle along its diagonal is 44 square inches. The area of a triangle is given by half the product of its base and height. In this case, the base and height of each right triangle correspond to the length and width of the rectangle. Therefore, we can write the equation for the area of one triangle as:

(1/2) * l * w = 44

Now we have a system of two equations with two variables:

w = l - 3
(1/2) * l * w = 44

Now, let's dive into solving this system of equations. The goal is to find the values of 'l' (length) and 'w' (width) that satisfy both equations simultaneously. We can use a method called substitution to achieve this. Since we already have an equation expressing 'w' in terms of 'l' (w = l - 3), we can substitute this expression for 'w' into the second equation:

(1/2) * l * (l - 3) = 44

Now, we have a single equation with only one variable, 'l'. Let's simplify and solve for 'l':

l * (l - 3) = 88  (Multiply both sides by 2)
l^2 - 3l = 88      (Distribute l)
l^2 - 3l - 88 = 0   (Rearrange into a quadratic equation)

We now have a quadratic equation in the standard form (ax^2 + bx + c = 0). To solve for 'l', we can use factoring, the quadratic formula, or completing the square. In this case, factoring seems like a straightforward approach. We need to find two numbers that multiply to -88 and add up to -3. These numbers are -11 and 8. So, we can factor the quadratic equation as follows:

(l - 11)(l + 8) = 0

This gives us two possible solutions for 'l':

l - 11 = 0  =>  l = 11
l + 8 = 0   =>  l = -8

Since the length of a physical object cannot be negative, we discard the solution l = -8. Therefore, the length of the rectangle is 11 inches. Now that we have the length, we can easily find the width using the equation w = l - 3:

w = 11 - 3 = 8

So, the width of the rectangle is 8 inches.

Before we proceed, let's verify that our solution for length and width is correct. We can plug the values l = 11 inches and w = 8 inches back into the original equations:

w = l - 3
8 = 11 - 3  (True)

(1/2) * l * w = 44
(1/2) * 11 * 8 = 44
44 = 44 (True)

Both equations hold true, confirming that our solution is accurate. The length of the rectangle is indeed 11 inches, and the width is 8 inches.

Now that we have determined the dimensions of the original rectangle, let's shift our focus to the right triangles formed by cutting the rectangle along its diagonal. Recall that each of these triangles has an area of 44 square inches.

We know the base and height of each right triangle correspond to the length and width of the rectangle, which are 11 inches and 8 inches, respectively. To fully characterize the triangle, we need to find the length of the hypotenuse, which is the diagonal of the rectangle. We can use the Pythagorean theorem for this purpose.

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 (the base and the height). Mathematically, this is expressed as:

a^2 + b^2 = c^2

where 'a' and 'b' are the lengths of the two shorter sides (base and height), and 'c' is the length of the hypotenuse.

In our case, a = 8 inches (width) and b = 11 inches (length). Let 'c' be the length of the hypotenuse (the diagonal). Plugging these values into the Pythagorean theorem, we get:

8^2 + 11^2 = c^2
64 + 121 = c^2
185 = c^2

To find 'c', we take the square root of both sides:

c = √185

So, the length of the hypotenuse (the diagonal) is √185 inches. We can approximate this value to get a better sense of its magnitude:

c ≈ 13.6 inches

Now that we have a thorough understanding of the rectangle's dimensions and the properties of the resulting triangles, we can address the core of the problem: identifying the correct statements about these geometric figures. Let's consider some potential statements and evaluate their truthfulness based on our findings.

  • Statement 1: The length of the rectangle is 11 inches.

    • Truthfulness: This statement is true. We calculated the length to be 11 inches.

  • Statement 2: The width of the rectangle is 14 inches.

    • Truthfulness: This statement is false. We found the width to be 8 inches.

  • Statement 3: The diagonal of the rectangle is √185 inches.

    • Truthfulness: This statement is true. We calculated the diagonal (hypotenuse) to be √185 inches.

  • Statement 4: The perimeter of each triangle is 44 inches.

    • Truthfulness: This statement is false. The perimeter of each triangle is the sum of its sides: 11 + 8 + √185 ≈ 11 + 8 + 13.6 ≈ 32.6 inches.

  • Statement 5: The area of the rectangle is 88 square inches.

    • Truthfulness: This statement is true. The area of the rectangle is length * width = 11 * 8 = 88 square inches.

  • Statement 6: The area of the rectangle is 44 square inches.

    • Truthfulness: This statement is false. The area of the rectangle is length * width = 11 * 8 = 88 square inches. The area of each triangle is 44 square inches.

Therefore, the three correct statements are:

  • The length of the rectangle is 11 inches.
  • The diagonal of the rectangle is √185 inches.
  • The area of the rectangle is 88 square inches.

In this article, we successfully dissected a geometric problem involving a rectangular piece of paper cut along its diagonal. By setting up and solving algebraic equations, applying the Pythagorean theorem, and carefully analyzing the given information, we were able to determine the dimensions of the rectangle and the properties of the resulting right triangles. We identified the true statements that accurately describe these geometric figures, showcasing the power of mathematical reasoning and problem-solving skills.