Heights & Distances

Introduction

• Heights and distances problems use trigonometric ratios to find lengths that are not directly measurable.
• Assumption: The line of sight, horizontal line, and vertical objects form right-angled triangles.

Key Definitions

Angle of Elevation

• When an object is above the observer’s eye level.
• The angle between the line of sight and the horizontal through the observer.

Angle of Depression

• When an object is below the observer’s eye level.
• The angle between the horizontal and the line of sight, measured downward.
• Angle of elevation = Angle of depression (alternate interior angles).

Bearing

• Direction of an object measured from a fixed reference direction.
• Cardinal directions: North (N), South (S), East (E), West (W).
• Bearing is measured as the angle between the line of sight and a cardinal direction.
• Examples:
  • NE = 45° from North towards East
  • ENE = 22½° from East towards North

Standard Trigonometric Relations Used

• sin θ = (Perpendicular) / (Hypotenuse)
• cos θ = (Base) / (Hypotenuse)
• tan θ = (Perpendicular) / (Base)
• cot θ = 1 / tan θ

Problems Based on Dimensions

Two-Dimensional Problems

• All points lie in the same vertical plane.
• Directions shown using a vertical cross (N–S, E–W).

Three-Dimensional Problems

• Objects lie in different vertical planes.
• Directions represented using an oblique cross for the horizontal plane.
• Common in advanced JEE Main word problems.

Useful Results

• m–n Theorem: In a triangle, (m + n) cotθ = m cotα − n cotβ
• If DE ∥ AB in ΔABC, then AB / DE = BC / DC

Shadow Problems

• Shadow of an object depends on the position of the light source.
• Join the top of the object to the light source.
• Extend the projection on the horizontal plane to find shadow length.

Circle-Based Geometry Used in Heights & Distances

• Angles subtended by the same chord at the circumference are equal.
• If points A, B, P, Q are concyclic, then ∠APB = ∠AQB
• Angle between a tangent and a chord equals the angle in the alternate segment.

Important Geometry Theorems

• Apollonius Theorem: If AD is the median of ΔABC, then AB² + AC² = 2(AD² + BD²)
• Angle Bisector Theorem: If AD bisects ∠A, then BD / DC = AB / AC
• Power of a Point: PA · PB = PT² (tangent case)
  PA · PB = PC · PD (secant case)

JEE Main Tips

• Always draw a neat diagram before forming equations.
• Clearly mark angles of elevation/depression.
• Check whether the problem is 2D or 3D before solving.
• Bearings are frequently tested in integer-angle form (30°, 45°, 60°).