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Mathematics

CD = 100 m, ∠ADB = 15° and ∠BDC = 45°

CD = 100 m, ∠ADB = 15° and ∠BDC = 45°. Concise Mathematics Solutions ICSE Class 10.

Statement (1): AB = tan 60° - tan 45°100\dfrac{\text{tan 60° - tan 45°}}{100}

Statement (2): AB = (100 tan 60° - 100 tan 45°) m.

  1. Both the statements are true.

  2. Both the statements are false.

  3. Statement 1 is true, and statement 2 is false.

  4. Statement 1 is false, and statement 2 is true.

Heights & Distances

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Answer

From figure,

In triangle BDC,

⇒tan 45°=BCDC⇒tan 45°=BC100⇒BC=100 tan 45°In triangle ADC,⇒tan 60°=ACDC⇒tan 60°=AB+BCDC⇒tan 60°=AB+100 tan 45°100⇒100 tan 60°=AB+100 tan 45°⇒AB=100 tan 60°−100 tan 45°\Rightarrow \text{tan 45°} = \dfrac{BC}{DC}\\[1em] \Rightarrow \text{tan 45°} = \dfrac{BC}{100}\\[1em] \Rightarrow BC = 100\text{ tan 45°}\\[1em] \text{In triangle ADC,}\\[1em] \Rightarrow \text{tan 60°} = \dfrac{AC}{DC}\\[1em] \Rightarrow \text{tan 60°} = \dfrac{AB + BC}{DC}\\[1em] \Rightarrow \text{tan 60°} = \dfrac{AB + 100\text{ tan 45°}}{100}\\[1em] \Rightarrow 100\text{ tan 60°} = AB + 100\text{ tan 45°}\\[1em] \Rightarrow AB = 100\text{ tan 60°} - 100\text{ tan 45°}

∴ Statement 1 is false, and statement 2 is true.

Hence, option 4 is the correct option.

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