From the top of an 18 m high building, the angles of depression of the top and the bottom of a tree are 30° and 60°. Find the height of the tree.

1. 1 · Write what is given and what is to be found Given: building AB = 18 m (A top, B foot). Angles of depression of the top C and the bottom D of tree CD are 30° and 60°. To find: CD.
2. 2 · Draw the figure Building AB, tree CD, dotted horizontal line at A. Draw CE horizontal from C to meet AB at E. Mark 60° at D and 30° at C (alternate angles). Mark the right angles at B and E.
3. 3 · Write the formula tan θ = opposite/adjacent; tan 60° = √3 and tan 30° = 1/√3.
4. 4 · Triangle ABD: find the distance
   • tan 60° = AB/BD = 18/BD
   • BD = 18/√3 = 6√3 m
5. 5 · Triangle AEC (CE = BD)
   • tan 30° = AE/CE = AE/(6√3)
   • AE = 6√3 × (1/√3) = 6 m
6. 6 · Find the height of the tree CD = EB = AB − AE = 18 − 6 = 12 m
7. 7 · Does it make sense? A 12 m tree is shorter than the 18 m building, which matches the figure. Many coconut trees are about this tall.
8. 8 · Check and box the answer Height of the tree = 12 m

Scoring tip: Find the shared horizontal distance first, from the angle at the foot. Then the second triangle gives the rest.