**Q) **A boy is standing on the top of light house. He observed that boat P and boat Q are approaching the light house from opposite directions. He finds that angle of depression of boat P is 45° and angle of depression of boat Q is 30°. He also knows that height of the light house is 100 m.

Based on the above information, answer the following questions.

(i) What is the measure of ∠APD?

(ii) If ∠YAQ = 30°, then ∠AQD is also 30°, Why?

(iii) Find length of PD

OR

**Find length of DQ**

**Ans: **

**(i) Measure of ∠APD:**

Given that ∠XAP = 45°

Since ∠XAD = 90°, therefore ∠PAD = ∠XAD – ∠XAP = 90° – 45° = 45°

In Δ APD, ∠APD + ∠PAD + ∠ADP = 180°

Since, ∠PAD = 45° and ∠ADP = 90°

∴ ∠APD = 180° – 90° – 45° = **45°**

**Therefore, the value of ∠APD is 45°**

**(ii) Reason of why ∠AQD is also 30°, If ∠YAQ = 30°:**

Since XY ǁ PQ and AQ cuts these parallel lines,

**therefore, ∠YAQ = ∠AQD = 30° because these are alternate interior angles.**

**(iii) Length of PD:**

In Δ APD, tan ∠APD = tan 45° =

1 =

PD = 100

**Therefore, length of PD is 100 m**

**OR**

**(iii) Length of DQ:**

In Δ AQD, tan ∠AQD = tan 30° =

PD = 100√3

**Therefore, length of DQ is 100√3.**