Question

Suppose you want to draw a rectangle where the width is 1 inch more than the length and the diagonal is 2 inches longer than the length. What are the dimensions o the rectangle?

Answer 1

To determine the dimensions of the rectangle with specific properties related to its width and diagonal, the Pythagorean Theorem is used. After setting up and solving the equation, it is found that the rectangle's dimensions are 3 inches in length, 4 inches in width, and 5 inches diagonally.

Step-by-step explanation:

To find the dimensions of a rectangle where the width is 1 inch more than the length and the diagonal is 2 inches longer than the length, we must set up equations based on the Pythagorean Theorem as follows:

Let the length be L inches.

Then, the width will be L + 1 inches.

The diagonal will be L + 2 inches.

According to the Pythagorean Theorem, L2 + (L + 1)2 = (L + 2)2

We expand and simplify this equation to find the value of L:

L2 + L2 + 2L + 1 = L2 + 4L + 4

Combining like terms, we get 2L2 + 2L + 1 = L2 + 4L + 4

Subtract L2 + 4L + 4 from both sides, we get L2 - 2L - 3 = 0

Factoring the quadratic equation, (L - 3)(L + 1) = 0, giving us the solutions L = 3 or L = -1. Since a negative length does not make sense in this context, we discard L = -1.

So, the dimensions of the rectangle are:

Length = 3 inches

Width = 3 + 1 = 4 inches

Diagonal = 3 + 2 = 5 inches

Answer 2

The dimensions of the rectangle are:

Length: 1 inch

Width: 2 inches

Step-by-step explanation:

Define variables: Let L be the length of the rectangle and W be the width. We know:

W = L + 1 (width is 1 inch more than the length)

Diagonal = L + 2 (diagonal is 2 inches longer than the length)

Use the Pythagorean Theorem: The diagonal and the sides form a right triangle. Applying the Pythagorean theorem:

(L + 2)^2 = L^2 + (L + 1)^2

Solve the equation: Expanding the squares and rearranging:

L^2 + 4L + 4 = L^2 + 2L + 1

2L = 3

L = 1.5

Find the width: Substitute L back into the first equation:

W = 1.5 + 1 = 2.5

Simplify: Since the dimensions are typically given in whole inches, round L to 1 and W to 2.

Therefore, the rectangle's dimensions are 1 inch for the length and 2 inches for the width.