A rectangular piece of paper has a width that is 3 inches less than its length. It is cut in half along a diagonal to create two congruent right triangles with areas of 44 square inches. Which statements are true? Check all that apply.

Question

The area of the rectangle is 88 square inches.
The equation x(x – 3) = 44 can be used to solve for the dimensions of the triangle.
The equation x2 – 3x – 88 = 0 can be used to solve for the length of the rectangle.
The triangle has a base of 11 inches and a height of 8 inches.
The rectangle has a width of 4 inches.

Answer 1

Let's define the variables first:

• Let \( l \) be the length of the rectangle.
• Let \( w \) be the width of the rectangle.

According to the problem, we have the following relationships:

1. The width is 3 inches less than the length: \[ w = l - 3 \]

2. The area of the rectangle is given by: \[ \text{Area} = l \times w \] Since the area of each triangle created by cutting the rectangle in half (along the diagonal) is 44 square inches, the area of the rectangle is: \[ \text{Area of rectangle} = 2 \times \text{Area of triangle} = 2 \times 44 = 88 \text{ square inches} \]

From the area of the rectangle, we have: \[ l \times w = 88 \]

Now we can substitute the width into the area formula: \[ l \times (l - 3) = 88 \]

This simplifies to: \[ l^2 - 3l - 88 = 0 \]

Now we check each statement provided:

1. The area of the rectangle is 88 square inches.

   • This statement is True.

2. The equation x(x – 3) = 44 can be used to solve for the dimensions of the triangle.

   • The triangle's area is 44, but we derive the rectangle's area formula using \( l \) and \( w \), so this statement is False.

3. The equation x² – 3x – 88 = 0 can be used to solve for the length of the rectangle.

   • We derived this equation from the area of the rectangle, so this statement is True.

4. The triangle has a base of 11 inches and a height of 8 inches.

   • The area of a triangle is given by \(\frac{1}{2} \times \text{base} \times \text{height} = 44\). If we set the base as 11 inches, we find: \[ \frac{1}{2} \times 11 \times h = 44 \implies 11h = 88 \implies h = 8 \]
   • This statement is True.

5. The rectangle has a width of 4 inches.

   • We know \( w = l - 3 \). Given \( l \) from the quadratic formula solutions, if we find \( l \), we can calculate \( w \). Solving \( l^2 - 3l - 88 = 0 \) gives us: \[ l = \frac{3 \pm \sqrt{9 + 352}}{2} = \frac{3 \pm 19}{2} \]
   • This results in \( l = 11 \) or a negative value. Thus, \( w = 11 - 3 = 8 \).
   • This statement is False.

Summary of True Statements:

• The area of the rectangle is 88 square inches.
• The equation \( x^2 – 3x – 88 = 0 \) can be used to solve for the length of the rectangle.
• The triangle has a base of 11 inches and a height of 8 inches.