Unit 5: Height, Distance, and Analytical Reasoning

I. Orientation

This unit applies mathematical measurement and systematic arrangement methods to infer unknown quantities or positions. Height-and-distance problems are governed mainly by right-triangle trigonometry, while seating-arrangement problems depend on translating verbal conditions into fixed relative positions. Accuracy depends on identifying the reference point, preserving direction, and using every condition consistently.

• Governing principle: Convert the information in words into a diagram, equation, or position table before solving.
• Measurement convention: Use the same unit throughout; convert metres, centimetres, kilometres, and degrees where necessary.
• Angle convention: Angles of elevation or depression are measured from a horizontal line unless stated otherwise.
• Reference convention: In seating problems, determine whether positions are counted from a person, a direction, or a fixed seat.
• Logical convention: A direct condition fixes an exact relationship; a negative condition such as “A is not beside B” eliminates arrangements.
• Verification principle: Check both numerical dimensions and all stated relationships after obtaining an answer.

II. Problems Based on Height and Distance — Trigonometric Measurement

A. Problems based on height and distance

Height-and-distance problems determine an unknown height, horizontal distance, or angle by modelling the situation as a right triangle. The central relationship is between an observer, a vertical object, and the horizontal ground distance separating them.

• Basic model: Let (h) be the vertical height, (d) the horizontal distance, and (\theta) the angle of elevation from the observer to the top of the object.

tan(theta) = h / d
h = d tan(theta)
d = h / tan(theta)

• Meaning of symbols: (h) and (d) are measured in the same linear unit; (\theta) is an angle; (\tan\theta) is the tangent of that angle.
• Angle of elevation: This is the angle between the observer’s horizontal line of sight and the upward line of sight to an object. If the observer sees the top of a tower above eye level, the angle is one of elevation.
• Angle of depression: This is the angle between the observer’s horizontal line and a downward line of sight. Because horizontal lines are parallel, an angle of depression from a tower equals the corresponding angle of elevation from the object below.
• Observer’s eye height: If the observer’s eye is (e) metres above ground and the object’s total height is (H), the vertical difference is (H-e).

tan(theta) = (H - e) / d
H = e + d tan(theta)

• Right-triangle ratios: Tangent is preferred when the opposite and adjacent sides are involved. Sine and cosine apply when the hypotenuse is known.

sin(theta) = opposite / hypotenuse
cos(theta) = adjacent / hypotenuse
tan(theta) = opposite / adjacent

• Useful standard values: The most frequently used values are:

tan(30 degrees) = 1 / sqrt(3)
tan(45 degrees) = 1
tan(60 degrees) = sqrt(3)

• Two observation points: If two observers on the same straight line view the top of an object at angles (\alpha) and (\beta), their distances from the foot of the object can be represented as (h\cot\alpha) and (h\cot\beta), provided both are on the same side.

distance difference = h(cot(alpha) - cot(beta))

Here (h) is the object’s vertical height and (\alpha,\beta) are the respective angles.

• Worked example: A person stands (20) m from a tower and observes its top at (45^\circ). Taking eye height as negligible:

h = d tan(theta)
h = 20 x tan(45 degrees)
h = 20 x 1
h = 20 m

The tower is (20) m high.

• Diagram discipline: Mark the object’s foot, the observer, the horizontal distance, the vertical height, and the sightline. Do not treat the sloping sightline as the ground distance.
• Common assumption: Unless the question gives ground slope, assume level ground and a vertical object. If the observer and object are at different elevations, include that elevation difference in the vertical side.
• Unit and calculator checks: Use degree mode for angles stated in degrees. A height calculated in metres cannot be added directly to a distance given in centimetres without conversion.
• Analytical limitation: Trigonometric equations provide a valid result only when the selected triangle matches the physical arrangement. A wrong angle or incorrect side identification produces a consistent-looking but invalid answer.

III. Linear Seating Arrangement — Ordering People on a Row

A. Linear seating arrangement

A linear seating arrangement places people or objects in a single row, usually facing the same direction unless the problem states otherwise. The task is to assign each person a unique position while satisfying direct, relative, and negative conditions.

• Position model: Number seats from left to right as (1,2,3,\ldots,n). Each person occupies exactly one position, and no two people occupy the same seat.
• Immediate adjacency: “A sits beside B” means their positions differ by one.

|position(A) - position(B)| = 1

• Fixed gap: If two people have (k) persons between them, their seat positions differ by (k+1).

|position(A) - position(B)| = k + 1

For example, two people with three persons between them are four positions apart.

• Relative order: “A sits to the left of B” gives (position(A) < position(B)). It does not necessarily mean they are adjacent.

• End conditions: “A sits at an extreme end” restricts A to position (1) or (n). “A sits second from the left” fixes A at position (2).

• Block formation: If A is immediately left of B, treat “AB” as one block. For “C sits beside either A or B” within a connected group, first preserve the required internal order.

• Negative conditions: “D does not sit beside E” eliminates arrangements where their position difference is (1). Negative clues should be applied after stronger fixed placements where possible.

• Facing direction: If all people face north, their left and right match the observer’s left and right. If all face south, their personal left and right are reversed relative to the observer.

• Worked example: Five people A, B, C, D, and E sit in a row. A is at the extreme left, B sits immediately right of A, C is at the extreme right, and D sits immediately left of E.

Positions:  1   2   3   4   5
            A   B   _   D   C

Since C is fixed at position (5), the block DE must occupy positions (3) and (4). The arrangement is A-B-E-D-C only if D is immediately left of E? No: the block must be “D E,” so positions (3,4) are D,E, producing:

Positions:  1   2   3   4   5
            A   B   D   E   C

This illustrates why the stated internal order of a block must be preserved.

• Counting from an end: If a person is (r) places from the left in a row of (n), the position from the right is:

position from right = n - r + 1

For (n=8) and (r=3), the person is (8-3+1=6) places from the right.

• “Either-or” clues: “F sits either second or fifth” creates two branches. Examine each branch separately and discard the one that violates another condition.
• Chain reasoning: Conditions such as A left of B, B left of C, and C left of D establish (A<B<C<D). They may not determine exact seats, but they sharply reduce possibilities.
• Systematic method: Begin with fixed ends, exact positions, and inseparable blocks. Next place gap conditions and directional chains. Finally test adjacency exclusions and “either-or” statements.
• Analytical limitation: A row has a meaningful left and right end. Reversing the complete arrangement may produce a different answer whenever the clues refer explicitly to left, right, or an end.

IV. Circular Seating Arrangement — Relative Position Around a Circle

A. Circular seating arrangement

A circular seating arrangement places people at equal or unspecified positions around a table or circle. Since a circle has no natural first seat, the solution depends primarily on relative positions, facing direction, and clockwise or anticlockwise relationships.

• Reference seat: Fix one person at the top or at an arbitrary position. This removes rotational duplicates without changing the relative arrangement.
• Circular counting: Count seats around the circle in the specified direction. If (n) people sit around a circle, the position after moving (k) places clockwise is calculated modulo (n).

new position = ((current position + k - 1) mod n) + 1

The formula numbers positions from (1) to (n).

• Circular permutations: For (n) distinct people, arrangements around a circle, ignoring rotation, equal:

(n - 1)!

The reduction by one occurs because any one person can be fixed as the reference. For six people, the number is (5! = 120), before additional restrictions.

• Immediate neighbours: Each person has exactly two neighbours. “A sits beside B” means B occupies one of the two seats directly next to A.
• Opposite positions: In an even-sized circle, the person opposite a given person is (n/2) seats away. In an eight-person arrangement, the opposite seat is four positions away.
• Facing centre: When people face the centre, a person’s left-hand neighbour is found anticlockwise, and the right-hand neighbour is found clockwise, assuming the standard top-view convention.
• Facing outside: When people face away from the centre, the left-right interpretation reverses: a person’s left-hand neighbour is clockwise and the right-hand neighbour is anticlockwise.
• Clockwise and anticlockwise: “A sits second clockwise from B” means one seat lies between B and A in the clockwise direction. The direction must be followed from B’s viewpoint as defined by the problem.
• Opposed halves: Circular questions often require careful separation of two cases:
  1. Facing centre: Personal left and right are opposite to the directions of a person viewed from outside the circle.
  2. Facing outside: Personal left and right agree with the direction seen from the centre-side perspective.
• Worked example: Six people A, B, C, D, E, and F sit facing the centre. A is fixed at the top. B sits second clockwise from A, C sits immediately clockwise from B, and D sits opposite A.

Number the seats clockwise from A:

Seat:  1   2   3   4   5   6
       A   _   _   D   _   _

B is second clockwise from A, so one seat lies between them. Thus B is at seat (3). C immediately clockwise from B is at seat (4), but seat (4) is already D, so this interpretation conflicts with the clue that D is opposite A. Therefore the arrangement must be tested using the problem’s exact counting convention: if “second clockwise” means the second seat encountered, B is seat (3), and the clues are inconsistent. This demonstrates the need to distinguish “second seat from” from “one person between”; a valid question must use compatible conditions.

• Block method: If A sits immediately clockwise from B, treat the ordered pair (BA) as a block. The block’s orientation cannot be reversed unless the clue permits either direction.
• Negative restrictions: “E does not sit opposite F” excludes one specific pair of positions when the circle has an even number of seats. “G is not adjacent to H” excludes both neighbouring positions.
• Position table: Record each placement with a direction label rather than relying on a mental image.

Reference: A at seat 1
Clockwise: seat 2, seat 3, seat 4, ...
Anticlockwise: seat n, seat n-1, seat n-2, ...

• Verification: After filling the circle, check every person’s two neighbours, all opposite pairs, every clockwise distance, and each facing-direction condition.
• Analytical limitation: Rotating the entire circle does not create a new relative arrangement, but reflecting it may reverse clockwise and anticlockwise relationships. Reflection is therefore not automatically equivalent to the original arrangement.