To calculate the length of $DG$ in the cuboid, we need to use the Pythagorean theorem in three dimensions. Here's the breakdown:

Step 1: Understanding the diagram

The line $DG$ is the diagonal that connects $D$ (bottom-left corner at the back) to $G$ (top-right corner at the front). This is a 3D diagonal within the cuboid. To calculate $DG$, we will:

1. First find the diagonal of the base ($DB$), which lies on the bottom face.
2. Then use $DB$ along with the height of the cuboid to calculate $DG$.

Step 2: Find $DB$ (Diagonal of the base)

The base of the cuboid is a rectangle with dimensions $43 \, \text{cm}$ (length) and $24 \, \text{cm}$ (width). The diagonal $DB$ can be calculated using:

$DB = \sqrt{43^2 + 24^2}$

$DB = \sqrt{1849 + 576}$

$DB = \sqrt{2425} \approx 49.25 \, \text{cm}$

Step 3: Find $DG$ (3D diagonal)

Now, $DG$ can be calculated using $DB$ and the height of the cuboid ($21 \, \text{cm}$). Again, use the Pythagorean theorem:

$DG = \sqrt{DB^2 + \text{height}^2}$

Substitute $DB = 49.25$ and height = 21:

$DG = \sqrt{49.25^2 + 21^2}$

$DG = \sqrt{2425 + 441}$

$DG = \sqrt{2866} \approx 53.6 \, \text{cm}$

Final Answer:

The length of $DG$ is 53.6 cm (to 1 decimal place).

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Relative Questions:

1. How would you calculate the diagonal of a cuboid given all its dimensions?
2. Why is the Pythagorean theorem essential for solving 3D diagonal problems?
3. Can this approach be used for other 3D shapes like pyramids or cylinders?
4. What is the relationship between 2D and 3D diagonals in geometry?
5. How can understanding cuboids help in real-life applications, such as architecture?

Tip:

Always start by analyzing the geometric figure and breaking it into simpler parts, like triangles or rectangles, to apply the Pythagorean theorem effectively.