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To calculate the length of \(DG\) in this cuboid, we need to follow these steps:

### Step 1: Identify the coordinates of points \(D\) and \(G\)
- \(D\) is located at the bottom-left of the cuboid, and its coordinates are \((0, 0, 0)\) (since it’s at the origin).
- \(G\) is the center point on the diagonal from \(A\) to \(C\) in the top face of the cuboid. We can calculate the coordinates of \(G\) by averaging the positions of points \(A\) and \(C\).

We are given the following dimensions:
- The length \(AB = 24\) cm.
- The width \(AD = 43\) cm.
- The height \(AE = 21\) cm.

### Step 2: Coordinates of Point \(G\)
Since \(G\) lies halfway along the diagonal from \(A\) to \(C\), we first find the coordinates of \(C\).

- \(A = (0, 0, 21)\)
- \(C = (43, 24, 21)\)

The midpoint \(G\) will have coordinates:
\[
G = \left( \frac{0 + 43}{2}, \frac{0 + 24}{2}, 21 \right) = (21.5, 12, 21)
\]

### Step 3: Use the 3D Distance Formula
Now, we use the distance formula in 3D to calculate the distance between \(D(0, 0, 0)\) and \(G(21.5, 12, 21)\). The distance formula is:
\[
DG = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
Substituting the coordinates:
\[
DG = \sqrt{(21.5 - 0)^2 + (12 - 0)^2 + (21 - 0)^2}
\]
\[
DG = \sqrt{21.5^2 + 12^2 + 21^2}
\]

### Step 4: Calculate the Result
Now, let's compute the length of \(DG\).The length of \(DG\) in the cuboid is \(32.4 \, \text{cm}\), rounded to 1 decimal place.

Would you like further details on this solution or have any questions?

Here are five related questions you might want to explore:
1. How do you calculate the length of a diagonal in a cuboid?
2. What is the formula for finding the diagonal of a rectangle or square face of a cuboid?
3. How do you find the volume of a cuboid?
4. Can the same method be used to find other diagonal lengths in different shapes?
5. How do 3D coordinate distances relate to real-world objects?

**Tip:** To find the diagonal of any rectangular face of a cuboid, apply the Pythagorean theorem to the lengths of its sides.

Work out the length of DG in this cuboid. Give your answer in centimetres (cm) to 1 d.p.

Learn how to calculate the diagonal DG in a cuboid with dimensions 43 cm, 24 cm, and 21 cm using the 3D distance formula. This geometry problem applies the Pythagorean theorem in 3D and is suitable for Grades 9-12 students.

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