Geometry: Pythagorean Theorem Word Problems - Free Printable

Let's solve the problem step by step using the information and setup provided in the image.

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Problem Statement:

- The length of a living room is 2 feet less than twice its width.
- The diagonal is 2 feet more than twice the width.
- We are to find the dimensions (width and length) of the room.

We'll use the Pythagorean Theorem, since we're dealing with a rectangle and its diagonal.

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Step 1: Define Variables

Let:
- $ x $ = width of the room (in feet)
- Then, length = $ 2x - 2 $ (since it's 2 feet less than twice the width)
- Diagonal = $ 2x + 2 $ (since it's 2 feet more than twice the width)

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Step 2: Apply the Pythagorean Theorem

For a rectangle:
$$
\text{width}^2 + \text{length}^2 = \text{diagonal}^2
$$

Substitute:
$$
x^2 + (2x - 2)^2 = (2x + 2)^2
$$

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Step 3: Expand Each Term

First, expand $ (2x - 2)^2 $:
$$
(2x - 2)^2 = 4x^2 - 8x + 4
$$

Next, expand $ (2x + 2)^2 $:
$$
(2x + 2)^2 = 4x^2 + 8x + 4
$$

Now substitute into the equation:
$$
x^2 + (4x^2 - 8x + 4) = 4x^2 + 8x + 4
$$

Simplify left side:
$$
x^2 + 4x^2 - 8x + 4 = 5x^2 - 8x + 4
$$

So:
$$
5x^2 - 8x + 4 = 4x^2 + 8x + 4
$$

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Step 4: Solve the Equation

Subtract $ 4x^2 + 8x + 4 $ from both sides:
$$
(5x^2 - 8x + 4) - (4x^2 + 8x + 4) = 0
$$

$$
x^2 - 16x = 0
$$

Factor:
$$
x(x - 16) = 0
$$

So:
$$
x = 0 \quad \text{or} \quad x = 16
$$

Since width can't be 0, we take:
$$
x = 16
$$

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Step 5: Find Dimensions

- Width = $ x = 16 $ feet
- Length = $ 2x - 2 = 2(16) - 2 = 32 - 2 = 30 $ feet
- Diagonal = $ 2x + 2 = 2(16) + 2 = 32 + 2 = 34 $ feet

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Step 6: Verify Using Pythagorean Theorem

Check:
$$
16^2 + 30^2 = 256 + 900 = 1156
$$
$$
34^2 = 1156
$$

✔ It checks out!

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✔ Final Answer:

- Width: 16 feet
- Length: 30 feet
- Diagonal: 34 feet

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Note on the Handwritten Work in Image:

The handwritten steps show:
$$
x^2 + (2x - 2)^2 = (2x + 2)^2
$$
Then incorrectly simplifies to:
$$
x^2 = (2x + 2)^2 - (2x - 2)^2
$$
Which is correct algebraically, but then the expansion is done incorrectly.

Let’s verify that approach:

$$
(2x+2)^2 - (2x-2)^2 = [4x^2 + 8x + 4] - [4x^2 - 8x + 4] = 16x
$$

So:
$$
x^2 = 16x \Rightarrow x^2 - 16x = 0 \Rightarrow x(x - 16) = 0
$$

Same result! So even though the handwriting is messy, the idea is sound — just needs careful expansion.

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🟩 Conclusion:

The dimensions of the room are 16 feet (width) by 30 feet (length).