The sightline, and the horizontal it gets measured against

What to assume they know

• The six trigonometric ratios of an acute angle in a right triangle, and which side each pair names — carried straight in from Chapter 8, which this chapter opens by pointing back to
• The exact values of the ratios at 30°, 45° and 60°
• That a level line and a plumb line meet at a right angle
• Reading a lettered figure: naming a segment by its two endpoints, and naming an angle by three letters with the vertex in the middle
• Solving a one-unknown linear equation of the form (unknown)/(number) = value
• Surds: leaving an answer as a multiple of √3 and, separately, turning it into a decimal with a stated approximation

What they should be able to do

• Identify, in a described or drawn situation, the observer's eye, the point of the object being aimed at, and the ray joining them
• Draw the level ray through the eye and mark the angle the sightline opens against it
• Explain why the level ray is taken through the eye rather than through the observer's feet, and what breaks if it is not
• Justify the right angle in the resulting triangle from the fact that the object stands plumb and the drawn line is level
• List the three field measurements the chapter says are needed before the height of a distant object can be computed
• Split the object's total height into the part above eye level and the part at or below it, and state which piece the triangle delivers
• Show that the lower piece equals the observer's height, using the rectangle formed by the two level lines and the two plumb lines
• Decide, for a given problem, whether an eye-height correction is required, and compute the height both with and without it

Where it usually goes wrong

• "The angle is measured up from the ground." It is measured from the level ray at the eye. Measuring from the ground would put the vertex at the observer's feet and change the triangle — and it is not what an instrument held to the eye reports.
• "The angle sits at the object." The angle in Fig. 9.1 is drawn at A, the eye. The angle at C, up at the minar's top, is the complement of it and is a different number.
• "The vertical side of the triangle is the height of the object." It is the height above eye level. Fig. 9.1 draws B partway up the minar for exactly this reason.
• "You can skip adding your height; it's rounding error." It is 1.5 m out of 30 m in Example 3, and it can be half the answer on an indoor measurement. It is never rounding error; it is a whole segment of the figure.
• "But Example 1 didn't add anything, so the rule is optional." Example 1 sights at ground level. There is nothing above the ground to add. The rule is uniform; the correction happens to be zero.
• "The distance to the foot is the distance to the object." DE is the ground run; AC is the slant. They differ by more than students expect at 60°, where the slant is twice the run.
• "The drawing has to be to scale for the reasoning to hold." The observer in Fig. 9.1 is drawn as a speck against the minar. The lettering carries the argument; the artwork only makes the scene recognisable.

Questions to check understanding

• Given a scene in words, draw and letter the figure, mark the sightline and the angle, and say where the right angle is
• Compute the height of a vertical object from a stated ground distance and a stated elevation of 30°, 45° or 60°
• The same, with an observer height supplied, so the answer requires the final addition — the standard one-mark trap
• Given an answer and a stated observer height, recover the height above eye level
• Explain in one or two lines why the horizontal is taken at the eye
• State which trigonometric ratio links two named sides of a lettered figure, without computing anything

Examples worth working on the board

Values marked verified are worked out here on data printed inside pp. 133–143; the chapter prints no answer key of its own.

• Fig. 9.1 (p. 133), two panels side by side. Left panel: a tall tapering minar drawn in blue; at the far left a very small standing figure marks the observer — I opened the printed page to confirm the figure is drawn, because it is barely a speck at page scale. Lettering: A the eye, E the feet, C the top of the minar, D the foot of the minar, B the point where the level ray from A crosses the minar's plumb line. Three dotted lines are drawn: A→C captioned as the sightline, A→B level, and E→D level along the ground. An arrow at A points into the angle between AC and AB and captions it. Right panel: the identical lettering with the minar's artwork removed, leaving triangle ABC sitting on rectangle ABDE. This pairing is the whole point of the figure.
• The height decomposition (§9.1, p. 135). CD = CB + BD, and BD = AE. So the triangle supplies CB only; AE is measured, not computed.
• The three required measurements (§9.1, pp. 134–135), stated as a numbered list: the ground run DE from the observer to the minar's foot, the angle at A, and the observer's height AE.
• The ratio choice for this figure (§9.1, p. 135). With ∠A known and AB measured, BC is reached through tan A = BC/AB, or equivalently through cot A = AB/BC. The page offers the pair and says either serves.
• Example 1 (p. 135, Fig. 9.4). Inputs: a vertical tower; the sighting is taken at ground level, 15 m out from the tower's foot; the elevation read there is 60°. Fig. 9.4 letters the tower's top A, its foot B, the ground point C, marks 60° at C and marks the 15 m along CB with a double arrow. Verified: AB = 15 tan 60° = 15√3 m ≈ 25.98 m. No eye-height term appears, and the reason is structural — the sighting is taken at ground level, so AE is zero here.
• Example 3 (p. 137, Fig. 9.6). Inputs: the observer is 1.5 m tall and stands 28.5 m from a chimney; the elevation of the chimney's top from her eyes is 45°. Fig. 9.6 letters the chimney AB with A on top, the observer CD with C at her feet and D at her eye, E on the chimney at eye height, and marks 45° at D. Verified: DE = CB = 28.5 m; AE = 28.5 tan 45° = 28.5 m; total height AB = 28.5 + 1.5 = 30 m. The correction is 1.5 m in 30 m, i.e. 5% of the answer — small, and not zero.
• A contrast worth building (mine, from the chapter's own two examples). Run Example 3's arithmetic with the correction dropped and the chimney comes out 28.5 m. Now shrink the scene: sight a 3 m ceiling fitting from 1.5 m away at 45° and the triangle gives 1.5 m, so the observer's height is half the answer. The correction is not fussiness that only matters for tall things; it matters most for short ones.
• Exercise 9.1 items that exercise this topic alone (pp. 141–142), inputs only: Q4 — the elevation of a tower's top read as 30° from a ground point 30 m from its foot. Q1 (Fig. 9.11) — a taut rope 20 m long from a vertical pole's top down to the ground, meeting the ground at 30°; the pole's height wanted. Q5 — a kite 60 m above the ground on a slack-free string inclined at 60°. Verified, working added here: Q4 gives 30 tan 30° = 10√3 ≈ 17.32 m; Q1 gives 20 sin 30° = 10 m; Q5 gives 60/sin 60° = 40√3 ≈ 69.28 m. Two items of the set pin the eye-height correction explicitly rather than leaving it implied: Q6, which gives the boy a height of 1.5 m against a 30 m building, and Q14 on p. 142, which gives a girl a height of 1.2 m and puts the balloon 88.2 m above the ground. Both need the observer's height taken off before the triangle can be used; Q14 belongs to Two angles in one figure, and the pair of equations they hand you, which handles it there.

Figures to have open

• Fig. 9.1 in both panels (p. 133). Not optional and not reducible to one panel: an added argument in sections 4–5 is that the right panel is the left panel with the scenery deleted. Redraw as a schematic — a tapering tower silhouette will do — rather than reproducing the printed artwork.
• Fig. 9.4 (p. 135): tower AB, ground point C, the 15 m run marked, 60° at C. Standard schematic.
• Fig. 9.6 (p. 137): chimney AB, observer CD, the eye-level foot E on the chimney, 45° at D, with BE and CD marked equal. Standard schematic; the equal marks are the point of the figure.
• A purpose-built overlay for section 3: one observer, two candidate horizontals (feet and eye), the two different angles they produce against the same sightline. This is an added figure; the book draws only the eye-level one.

Where this sits in the book

• NCERT Class X Mathematics, Chapter 9 "Some Applications of Trigonometry", §9.1 "Heights and Distances", pp. 133–135 for the construction and pp. 135 and 137 for Examples 1 and 3.
• Fig. 9.1 (p. 133), Fig. 9.4 (p. 135), Fig. 9.6 (p. 137).
• Exercise 9.1 questions 1, 4, 5 and 6, pp. 141–142.
• §9.2 Summary, p. 143, item 1(i) and item 2.
• Cross-reference the book itself makes: the opening paragraph of §9.1 says Fig. 9.1 is a redrawing of a figure from Chapter 8, and points back to that chapter for the ratios. Chapter 8 is not in this chapter's page range and is cited here only as the book cites it.