We are given a rectangle DEFG, where the length of side DF is $5x - 5$ and the length of side EG is $x + 27$. Since DEFG is a rectangle, opposite sides are equal, meaning:

• DF = EG

This leads to the equation:

$5x - 5 = x + 27$

Let's solve for $x$:

$5x - x = 27 + 5$ $4x = 32$ $x = 8$

Now that we have $x = 8$, we can find the lengths of DF and EG:

$DF = 5x - 5 = 5(8) - 5 = 40 - 5 = 35$ $EG = x + 27 = 8 + 27 = 35$

Thus, both sides are 35, and the value of $x$ is 8.

To find the length of each diagonal, we use the Pythagorean theorem. In a rectangle, the diagonals are equal in length and can be calculated using the formula:

$\text{Diagonal} = \sqrt{DF^2 + EG^2}$

Substitute the values for DF and EG:

$\text{Diagonal} = \sqrt{35^2 + 35^2} = \sqrt{1225 + 1225} = \sqrt{2450} \approx 49.5$

So, the value of $x$ is 8, and the length of each diagonal is approximately 49.5.

Answer:

• $x = 8$
• DF = 35
• EG = 35
• Diagonal length = 49.5 (approx)

Would you like more details or clarification on any part of the solution?

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Here are some related questions:

1. How does the Pythagorean theorem apply to rectangles?
2. What is the general formula for the length of a diagonal in any rectangle?
3. Can you solve this problem if the sides of the rectangle are given in terms of different expressions?
4. How does the Pythagorean theorem generalize to 3D shapes like cuboids?
5. What other methods can be used to find the length of a diagonal in geometric shapes?

Tip: Always check if you can apply the Pythagorean theorem to solve for diagonal lengths in right-angled shapes like rectangles!