10 min readPathway sheet3-mark question, or a 5-mark long answer

Two heights over one distance

When one object stands on top of another or a building faces a tower, both triangles share the same horizontal distance. Find that distance first, then each height.

Shared sideThe horizontal distance d is common to both triangles

Flagstaff on a buildingFlagstaff = height to its top − height of the building

Tower seen from a building topTower height = building height + d × tan (elevation)

Depression of a footd = building height ÷ tan (depression)

How to draw the figure

Flagstaff or tower on a building: draw the building BC as a vertical line and the flagstaff CD on top of it, in the same line. Mark a point A on the ground. Join AC and AD and mark both angles at A.
Building facing a tower: draw a short vertical AB (building) and a tall vertical CD (tower) on the same ground line BC.
From the top A of the building, draw a dotted horizontal AE to the tower, meeting it at E. Then CE = AB and AE = BC.
Mark the angle of elevation of D above AE and the angle of depression of C below AE.
Write the given height and mark the right angles at B, C and E.

Method

1
Draw the figure and spot the common horizontal side.
2★
Use the triangle with two known values to find the common side d.
3
Use d in the second triangle to find the other height.
4
Add or subtract heights as the figure shows.
5
Use √3 = 1.732 and write units.

★ key step

A mobile tower on a building

Problem

A mobile tower stands on top of a 15 m tall building. From a point on the ground, the angles of elevation of the bottom and the top of the mobile tower are 30° and 60°. Find the height of the mobile tower.

Working

Let BC = 15 m be the building and CD the mobile tower. A is the point on the ground. ∠BAC = 30°, ∠BAD = 60°.
In ΔABC: tan 30° = BC/AB, so 1/√3 = 15/AB and AB = 15√3 m.
In ΔABD: tan 60° = BD/AB, so √3 = BD/(15√3).
BD = 15√3 × √3 = 15 × 3 = 45 m.
CD = BD − BC = 45 − 15 = 30 m.

Answer

The mobile tower is 30 m tall.

A building facing a tower

Problem

A building is 12 m high. From its top, the angle of elevation of the top of a tower is 60°, and the angle of depression of its foot is 30°. Find the height of the tower and the distance between the building and the tower. (Use √3 = 1.732)

Working

Let AB = 12 m be the building and CD the tower. Draw AE ⟂ CD, so AE = BC and CE = AB = 12 m.
Angle of depression of C from A is 30°, so ∠ACB = 30° (alternate angles).
In ΔABC: tan 30° = AB/BC, so 1/√3 = 12/BC and BC = 12√3 m.
In ΔAED: tan 60° = DE/AE, so √3 = DE/(12√3) and DE = 36 m.
CD = CE + DE = 12 + 36 = 48 m.
BC = 12 × 1.732 = 20.784 ≈ 20.78 m.

Answer

Tower = 48 m; distance between them ≈ 20.78 m.

Flagstaff on a building vs building facing a tower

Flagstaff on a buildingBoth angles are measured at one point on the ground. The common side is the ground distance to the building. Flagstaff = top height − building height.

Building facing a towerBoth angles are measured at the top of the building. The common side is the horizontal AE, equal to the gap between them. Tower = building height + DE.

In real life

A mobile tower on a rooftop or a flag on a fort's rampart is a height standing on a height. Telecom engineers use two angle readings from the street to check how tall the tower is.

Exam angle

3 marks: “A 10 m flagstaff stands on a building. From a point on the ground, the angles of elevation of the top and the bottom of the flagstaff are 60° and 45°. Find the height of the building. (Use √3 = 1.732)” Let the building be h. tan 45° gives d = h. tan 60° = (h + 10)/h, so √3 h = h + 10. Then h = 10/(√3 − 1) = 5(√3 + 1) = 5 × 2.732 = 13.66 m.

5 marks: “From the top of a 60 m high building, the angles of depression of the top and the bottom of a tower are 30° and 60°. Find the height of the tower and the distance between them. (Use √3 = 1.732)” Distance d = 60/tan 60° = 20√3 = 34.64 m. Drop from the building top to the tower top = d tan 30° = 20√3 × 1/√3 = 20 m. Tower = 60 − 20 = 40 m.

Remember it

Stacked heights share the ground distance. Find d first from the easier triangle.