Question

CD = 100 m, ∠ADB = 15° and ∠BDC = 45°

Statement (1): \[\mathrm{AB}=\frac{\tan 60^{\circ}-\tan 45^{\circ}}{100}\]

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

Options

• Both the statements are true.

• Both the statements are false.

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

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

Answer 1

Statement 1 is false, and statement 2 is true.

Explanation:

From figure,

In triangle BDC,

\[\Rightarrow \tan 45^\circ = \frac{BC}{DC}\]

\[\Rightarrow \tan 45^\circ = \frac{BC}{100}\]

⇒ BC = 100 tan 45°

In triangle ADC,

\[\Rightarrow \tan 60^\circ = \frac{AC}{DC}\]

\[\Rightarrow \tan 60^\circ = \frac{AB + BC}{DC}\]

\[\Rightarrow \tan 60^\circ = \frac{AB + 100 \tan 45^\circ}{100}\]

⇒ 100 tan 60° = AB + 100 tan 45°

⇒ AB = 100 tan 60° − 100 tan 45°

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