Height of lighthouse based on angle difference I have a question in my maths book: A lookout in a lighthouse tower can see two ships approaching the coast. Their angles of depression are 25° and 30°. If the ships are 100 m apart, show that the height of the lighthouse, to the nearest metre, is 242 metres. I have no clue how to solve it, please can someone give me a step-by-step process of how to work it out??

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HERE LET AB=H (HEIGHT OF LIGHTHOUSE) Here lets apply trigonometry in $\triangle ABC$ $$tan\angle ACB=\frac{AB}{BC}$$ $$tan30=\frac{AB}{x}$$ $$\rightarrow AB=H=tan30(x)\tag1$$

Applying the same thing in $\triangle ABD$ we can get $$tan\angle ADB=\frac{AB}{100+x}$$ $$tan25=\frac{AB}{100+x}$$ $$\rightarrow AB=H=tan25(100+x)\tag2$$

As $(1)=(2)$ We can easily get $$x=420.09m\tag3$$ using $tan25=0.466$ and $tan30=0.577$

Now again coming back to $\triangle ABC$ and using equation $(1)$ $$H=420.09(tan30)$$ $$H=242.5m \sim 242m$$