CDS Mathematics · Heights and Distances

Towers on Plane Figures and Bearings

When towers stand at the corners or centre of a square, rectangle or hexagon, the ground distances come from the plane figure, and each tower is its own right triangle.

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Why this matters

Seven PYQs, five of them HARD — the hardest page in the chapter, because the geometry is in two planes. Find each ground distance from the figure first (a side, a diagonal, half a diagonal), then use the elevation for each tower separately.

Concept 1 of 1: Ground distances from the figure

The towers stand upright, so every line of sight is still a right triangle — but its ground leg now runs across a square, a rectangle or a hexagon. Solve the flat figure first, then each tower.

Definition

Regular hexagon of side $s$ , from vertex $A$ : $AB = s$ , $AC = s\sqrt3$ , $AD = 2s$ .
Square of side $l$ : centre to corner $= \dfrac{l}{\sqrt2}$ .
Tower at a corner of a rectangle: the two sides and the diagonal are the three ground distances.
Bearings: $N\,\theta\,E$ is $\theta$ east of north; put the bank on one axis and write each bearing as a tangent.

Hexagon diagonals

AB : AC : AD = 1 : \sqrt3 : 2

Worked example

Poles stand at

B

and

D

of a regular hexagon

ABCDEF

. From

A

, both tops are at elevation

45^\circ

. Find the ratio of the poles.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q60Hard

Example 1 · Heights and Distances · Towers on Plane Figures and Bearings

A vertical tower standing at the corner of a rectangular field subtends angles of

60^\circ

and

45^\circ

at the two nearer corners. If

\theta

is the angle that the tower subtends at the farthest corner, then what is

\cot\theta

equal to?

Which diagonal of the hexagon?

From

A

, the next-but-one vertex

C

s\sqrt3

away and the opposite vertex

D

2s

. Using

2s

for

C

changes the ratio of the towers.

Drill 6 more on ground distances from the figure

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Ground distances from the figure
Hexagon diagonals
$AB : AC : AD = 1 : \sqrt3 : 2$

Watch out for (1)

Which diagonal of the hexagon?
→ Ground distances from the figure

Test yourself on Heights and Distances

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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