Looking up versus looking down: elevation and depression

What to assume they know

• The sightline, and the horizontal it gets measured against — the sightline, the level ray through the eye, and the right triangle they produce
• Alternate angles at a transversal cutting a pair of parallel lines, from the earlier work on lines and angles
• That opposite sides of a rectangle are equal, and that a rectangle's sides meet squarely
• Trigonometric ratios at 30°, 45° and 60°
• Reading a figure in which one lettered point serves two triangles at once

What they should be able to do

• Classify a described sighting as an elevation or a depression by asking only whether the target sits above or below the eye
• Draw the level ray in a depression figure, starting it at the observer and running it out over the object, so the depression angle is the wedge between that ray and the sightline
• Explain why the depression angle is drawn outside the triangle, while the elevation angle is drawn inside it
• Identify the sightline as a transversal of two parallel level rays, and name the alternate-angle pair it creates
• Deduce that a depression from a high point equals the elevation from the low point, and use that to relabel a figure before computing
• Locate, inside a two-building depression figure, the rectangle that makes the horizontal distances equal
• Convert any of the exercise set's depression problems into an equivalent elevation problem and say what was gained
• Justify why nothing in the method needs the observer and the object to be at comparable heights

Where it usually goes wrong

• "Depression is measured down from the vertical." It is measured down from the level ray, exactly as elevation is measured up from it. A 30° depression is a shallow look, not a steep one.
• "The depression angle is an angle of the triangle." In Fig. 9.9 the printed 30° is at P between PQ and PB, and PQ is not a side of triangle PBD. The angle has to be moved to B before any ratio can be written. Skipping this is the single commonest wrong start on these problems.
• "Depression needs its own formulas." It needs one relabelling. After that the same ratios apply.
• "Elevation and depression between the same two points are different." They are equal, and the parallel-lines argument is why. Students who take this on trust cannot reconstruct it under exam pressure; make them draw the two horizontals.
• "The two horizontals are parallel because they look parallel." They are parallel because both are level. That is what "level" means, and it is the only geometric input the whole chapter takes from outside trigonometry.
• "In a two-building problem the far building's height enters through its own triangle." In Example 6 the 8 m enters as DC, a piece of the tall building's plumb line, via the rectangle. The short building has no triangle of its own.

Questions to check understanding

• Given a described sighting, state whether it is an elevation or a depression and draw the level ray correctly
• Redraw a printed depression figure with the angle transferred to the far vertex, and justify the transfer in one line
• Two depressions from one elevated point to two objects on the same side — find the gap between the objects
• One elevation and one depression from the same rooftop — find a total height as a sum of two pieces
• A depression pair with a time interval, giving a speed or a further time
• Explain why a point's depression below an observer equals that observer's elevation above the point

Examples worth working on the board

Values marked verified are worked out here on data printed inside pp. 133–143.

• Fig. 9.2 (p. 134). A person stands on a level ground line at the left. A dotted level ray runs right from the eye and carries a caption naming it. A solid ray climbs to the upper right and ends at a small object, which is captioned. Two more captions sit in the wedge between the rays, one naming the sightline and one naming the angle. A small hill range is drawn under the level ray to indicate distance. Everything about the construction is the same as Fig. 9.1's, at a coarser grain.
• Fig. 9.3 (p. 134). A multi-storey building with a railed balcony fills the left of the frame; the observer stands at the balcony rail. A dotted level ray runs right from her, captioned. A solid sightline runs down to the lower right and ends at a small captioned object. The angle caption sits between the two rays, above the sightline. Note as a check: the depression angle is drawn in the open air outside any triangle, and that is exactly what makes it feel like a different animal from elevation. Section 4 exists to defuse this.
• The chapter's own open question (p. 134). Immediately after Fig. 9.3 the book points at a figure from Chapter 8 and asks the reader to sort its sightlines into elevations and depressions. It supplies no verdict. Is the target above the eye or below it?
• Example 6 (pp. 139–140, Fig. 9.9). Inputs: an 8 m building is sighted twice from the roof of a taller one — its own roof at a depression of 30°, its base at 45°. Wanted: the tall building's height and the gap between the two buildings. Fig. 9.9 letters the tall building PC with P on top and C at its foot, the short building AB with A at its foot and B on top, D the point of PC level with B, and Q a point on the level ray leaving P. Both angles are printed at P. The rest of the figure is a rectangle ABDC.
  • The step this topic is about (p. 139): PB cuts the parallel pair PQ and BD, so the angle at P and the angle at B are alternate and equal, giving ∠PBD = 30°. The same move at the lower level gives ∠PAC = 45°.
  • Verified, an added algebra: from ∠PBD = 30°, BD = PD√3. From ∠PAC = 45°, PC = AC. With AC = BD and DC = AB = 8, PD + 8 = PD√3, so PD = 8/(√3 − 1) = 4(√3 + 1). Height PC = PD + 8 = 4(3 + √3) ≈ 18.93 m, and the gap AC is the same number.
• Example 7 (pp. 140–141, Fig. 9.10). Inputs: standing on a bridge over a river, an observer sees the two banks at depressions of 30° and 45°; the deck sits 3 m above the banks. Wanted: the river's width. Fig. 9.10 letters A and B on the two banks, P the point on the bridge with both angles printed at it, and D the foot of the 3 m perpendicular from P, with the 3 m labelled on the drawing. Verified: the swap turns the depressions into ∠A = 30° and ∠B = 45°; then AD = 3/tan 30° = 3√3 and BD = 3; width AB = 3 + 3√3 = 3(1 + √3) ≈ 8.20 m. Note that the 45° branch needs no ratio table at all — the triangle is isosceles, so that leg simply repeats the 3 m.
• Exercise 9.1's depression items (pp. 142–143), inputs only. Q12: from a 7 m rooftop, a cable tower's top is at 60° elevation and its foot at 45° depression; the tower's height is wanted. Q13: from a 75 m lighthouse, two ships on the same side, one directly behind the other, are at depressions of 30° and 45°; the gap between them is wanted. Q15: a car approaching a tower at a steady speed is seen at 30° depression, then six seconds later at 60°; the further time to the tower's foot is wanted. Verified, working added here: Q12 gives 7(1 + √3) ≈ 19.12 m; Q13 gives 75(√3 − 1) ≈ 54.9 m; Q15 gives 3 s, and the tower's height cancels out of it entirely.
• The chapter's closing summary (p. 143) states elevation and depression as two separate items, (ii) and (iii) of point 1, phrased in parallel with each other. Worth showing as the last beat, because the parallel phrasing is itself the argument this topic makes.

Figures to have open

• Fig. 9.2 and Fig. 9.3 (p. 134), redrawn as clean schematics. Both are needed and they must be shown together — the topic's first claim is that they are the same drawing.
• Fig. 9.9 (p. 139): P, Q, B, D, A, C with both angles at P, the rectangle ABDC visible, and PB drawn through. This figure carries sections 5–7 and 10 on its own.
• Fig. 9.10 (p. 140): the bridge, P with both angles, the 3 m perpendicular down to D, and A and B on the two banks.
• A purpose-built two-panel movement for section 7: the same two points, the depression drawn from the upper one, then the figure rotated in place so the reader sees the elevation from the lower one is the same wedge. This is an added figure.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Class X Mathematics, Chapter 9 "Some Applications of Trigonometry", §9.1 "Heights and Distances", p. 134 for both definitions and the two figures, pp. 139–141 for Examples 6 and 7.
• Figs. 9.2 and 9.3 (p. 134), Fig. 9.9 (p. 139), Fig. 9.10 (p. 140).
• Exercise 9.1 questions 12, 13 and 15, pp. 142–143.
• §9.2 Summary, p. 143, point 1 items (ii) and (iii).
• Cross-references the book itself makes, both on p. 134: the depression definition is motivated by a figure from Chapter 8, and the reader is then asked to classify the sightlines in a second Chapter 8 figure. Neither figure is reprinted here, and the only figures in this chapter are numbered 9.1 to 9.13. Chapter 8 lies outside this chapter's page range and is cited only because the book cites it.