https://file.aiquickdraw.com/mathbot/file/1729263352988nfo0wg4m.jpg

Let's work through the multiple-choice questions based on the image you provided:

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**1. The distance of the point (-6, 8) from the x-axis is:**

- The distance from the x-axis is determined by the y-coordinate. 
- Here, the y-coordinate is 8, so the distance is **8 units**.

**Answer:** (c) 8 units

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**2. If the angle of depression of an object from a 75 m high tower is 30°, then the distance of the object from the tower is:**

- Using trigonometry, we apply the formula for the tangent of the angle of depression:
  
  \[
  \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{75}{d}
  \]
  \[
  d = \frac{75}{\tan(30^\circ)} = \frac{75}{\frac{1}{\sqrt{3}}} = 75 \times \sqrt{3} \approx 75 \times 1.732 \approx 129.9 \text{ m}
  \]

  The closest option is **150 m**.

**Answer:** (d) 150 m

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**3. If the height and length of the shadow of a man are equal, then the angle of elevation of the sun is:**

- If the height and the length of the shadow are equal, the angle of elevation forms a 45° angle (as the height and shadow length form an isosceles right triangle).

**Answer:** (a) 45°

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Would you like any further details or clarifications on any of these problems?

Here are some related questions for practice:

1. What is the distance of the point (4, -9) from the x-axis?
2. How would the distance change if the angle of depression is 45° from the tower?
3. If a man's shadow is twice his height, what would be the angle of elevation of the sun?
4. What is the distance of the point (-6, 8) from the origin?
5. How does the distance formula work in 3-dimensional space?

**Tip:** When working with angles of depression and elevation, always remember that they form right triangles with the horizontal line (ground level).

1. The distance of the point (-6, 8) from the x-axis is...
2. If the angle of depression of an object from a 75m high tower is 30°, then the distance of the object from the tower is...
3. If the height and length of the shadow of a man are equal, then the angle of elevation of the sun is...

Solve problems involving coordinate geometry and trigonometry. Learn how to calculate the distance of a point from the x-axis and use angles of depression and elevation to find distances. Suitable for students in Grades 9-11.

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