https://file.aiquickdraw.com/mathbot/file/1721869281576kqlqawyk.png

To find the area of \( \triangle DUO \), we need to determine the base and the height of the triangle.

1. **Base \( DU \)**: From the diagram, \( D \) is located 10 units from \( M \) which is located 4 units from \( O \). Therefore, the total horizontal distance from \( O \) to \( D \) is \( 4 + 10 = 14 \) units. However, we need the direct length of \( DU \).

2. **Height \( h \)**: The height \( h \) is the vertical distance from \( D \) to \( U \). \( DR \) is perpendicular to \( OP \), and from the given, the vertical distance \( DR = 4.5 \) units. Since \( DR \) is perpendicular and \( D \) is directly above \( R \), the distance \( DU = DR = 4.5 \) units.

Now, since \( DU \) is the height and \( DO \) (directly horizontally aligned) acts as the base, we calculate the area using the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is \( DO = 14 \) units and the height is \( 4.5 \) units.

Therefore,
\[
\text{Area} = \frac{1}{2} \times 14 \times 4.5 = 7 \times 4.5 = 31.5 \text{ square units}
\]

Thus, the area of \( \triangle DUO \) is \( 31.5 \) square units.

Would you like more details or have any questions about this solution?

Here are some other questions you might find interesting:
1. How do you find the area of a trapezoid?
2. What is the Pythagorean theorem and how is it applied?
3. Can you explain the difference between a rhombus and a square?
4. How do you calculate the area of a circle?
5. What are the properties of similar triangles?

**Tip:** When solving for areas in geometry, always identify and confirm the correct base and height perpendicular to each other.

Learn how to find the area of triangle DUO with a detailed step-by-step calculation. This problem involves determining the base and height of the triangle and using the area formula A = 0.5 * base * height. Suitable for students in Grades 7-9.

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