Square plot, pole at D — mixed observation points ABCD is a square. The angle of elevation of the top of a pole at D is 30° as seen from A and also from C. From B, the angle of elevation is Θ. Find tan Θ.

Introduction / Context:In a square, distances from D to adjacent corners (A and C) equal the side length s, while distance from D to the opposite corner (B) equals the diagonal s√2. Using the common pole height derived from the 30° observations at A and C, we can find tan Θ at B.

Given Data / Assumptions:

• AD = CD = s; BD = s√2.
• tan 30° = height/AD ⇒ height h = s/√3.
• We seek tan Θ = h/BD.

Concept / Approach:Use the 30° observation to compute h in terms of the side s, then evaluate tan Θ from the more distant point B using the diagonal baseline BD.

Step-by-Step Solution:

Verification / Alternative check:If s = √6 (arbitrary scaling), then h = 1 and BD = √12; tan Θ = 1/√12 = 1/ (2*√3) = 1/√6 after simplification consistency check.

Why Other Options Are Wrong:√6 or √3/√2 imply much larger angles inconsistent with the geometry; √2/√3 is also incorrect given h and BD.

Common Pitfalls:Treating BD as s instead of s√2, or mixing up sine and tangent when computing h.

Final Answer:1/√6