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Mathematics

The length of DC is :

  1. 15315\sqrt{3} cm

  2. 15(3−1)15(\sqrt{3} - 1) cm

  3. 15(3+1)15(\sqrt{3} + 1) cm

  4. 10310\sqrt{3} cm

The length of DC is : Heights and Distances, Concise Mathematics Solutions ICSE Class 10.

Heights & Distances

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Answer

We know that,

tan θ = PerpendicularBase\dfrac{\text{Perpendicular}}{\text{Base}}

From figure,

In △ ABC,

⇒ tan 60° = ABBC\dfrac{AB}{BC}

⇒ 3=15BC\sqrt{3} = \dfrac{15}{BC}

⇒ BC = 153\dfrac{15}{\sqrt{3}}

Rationalizing the denominator,

⇒BC =153×33⇒BC =1533⇒BC =53.\Rightarrow \text{BC } = \dfrac{15}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} \\[1em] \Rightarrow \text{BC } = \dfrac{15\sqrt{3}}{3} \\[1em] \Rightarrow \text{BC } = 5\sqrt{3}.

In △ ABD,

⇒ tan 30° = ABBD\dfrac{AB}{BD}

⇒ 13=15BD\dfrac{1}{\sqrt{3}} = \dfrac{15}{BD}

⇒ BD = 15315\sqrt{3} cm.

From figure,

DC = BD - BC = 153−53=10315\sqrt{3} - 5\sqrt{3} = 10\sqrt{3} cm.

Hence, Option 4 is the correct option.

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