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Mathematics

The length of AB is:

The length of AB is: Solution of Right Triangles, Concise Mathematics Solutions ICSE Class 9.

  1. (10 - 3\sqrt{3}) cm

  2. 10(3\sqrt{3} - 1) cm

  3. 10 3\sqrt{3} cm

  4. 10(3\sqrt{3} + 1) cm

Trigonometric Identities

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Answer

By formula,

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

In triangle BCD,

⇒tan B=CDBC⇒tan 45°=10BC⇒1=10BC⇒BC=10.\Rightarrow \text{tan B} = \dfrac{CD}{BC}\\[1em] \Rightarrow \text{tan 45°} = \dfrac{10}{BC}\\[1em] \Rightarrow 1 = \dfrac{10}{BC}\\[1em] \Rightarrow BC = 10.

In triangle ACD,

⇒tan A=CDAC⇒tan 30°=10AC⇒13=10AC⇒AC=103.\Rightarrow \text{tan A} = \dfrac{CD}{AC}\\[1em] \Rightarrow \text{tan 30°} = \dfrac{10}{AC}\\[1em] \Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{10}{AC}\\[1em] \Rightarrow AC = 10\sqrt{3}.

From figure,

⇒ AB = AC - BC

⇒ AB = 10310\sqrt{3} - 10 cm

⇒ AB = 10(3\sqrt{3} - 1) cm.

Hence, option 2 is the correct option.

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