Class 10 Maths · Chapter 9 · Topic 1 of 3
Start with the horizontal, then mark the angle
Measure sight angles from a horizontal line.
Ideas you’ll use
- Elevation and depression
- An angle of elevation is measured above the horizontal through the observer. An angle of depression is measured below that horizontal.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
Two parallel horizontals explain equal sight angles
y-coordinate (unit)
x-coordinate (unit)
Equal units on both axes.
- Lower horizontal · Original coordinate construction.
- Upper horizontal · Original coordinate construction.
- Sight line · Original coordinate construction.
- 1. Ground observer · (0, 0)
- 2. Balcony observer: 30° depression · (6.928, 4)
Assumptions: Both axes use the same length per unit. Straight segments join the stated coordinates. The algebra gives exact lengths.
Read the plotted values
Lower horizontal
| x-coordinate (unit) | y-coordinate (unit) |
|---|---|
| 0 | 0 |
| 8 | 0 |
Upper horizontal
| x-coordinate (unit) | y-coordinate (unit) |
|---|---|
| 0 | 4 |
| 8 | 4 |
Sight line
| x-coordinate (unit) | y-coordinate (unit) |
|---|---|
| 0 | 0 |
| 6.9282 | 4 |
The line rises 4 units over 4√3 units, so its angle to either horizontal is 30°. Equal scales on both axes.
Locate the observer’s eye
A line of sight joins the eye to the object. The reference line through the eye is horizontal.
Above horizontal: elevation; below: depressionIn the picture: Identify the sight line and horizontal reference.
Read all the reasoning together
Locate the observer’s eye
A line of sight joins the eye to the object. The reference line through the eye is horizontal.
Above horizontal: elevation; below: depressionUse parallel horizontals
The depression angle equals the elevation angle at the other endpoint on this same sight line.
depression 30° ⇒ elevation 30°Do not use the vertical reference
At a balcony, the angle between a downward sight line and a vertical wall is complementary.
90°−30°=60°
Measure sight angles from a horizontal line.
Worked example: Translate a downward view
Depression from a balcony = 30° ⇒ elevation to that balcony from the ground point = 30°This uses parallel horizontal lines. The angle inside the triangle at the balcony between the vertical wall and sight line is 60°, not 30°.
Read the supporting explanation
A line of sight joins the observer’s eye to the point being viewed. Do not measure the angle from a vertical wall. Parallel horizontals give equal alternate interior angles, so a depression angle equals the elevation angle along the same sight line.
Go deeper: Make the assumptions visible
Use level ground, vertical objects and the observer and object in the same vertical plane when stated. A sketch is a mathematical model; do not infer an unstated distance from its apparent size.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
A roof observer looks down at 40° below horizontal. What angle does that sight line make with the vertical?
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