If BD ⊥ AC and BC = 2√3cm, find the length of AD.

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In the adjoining figure, ABC is right-angled triangle at B and ABD is right angled triangle at A. If BD ⊥ AC and BC = 2√3cm, find the length of AD.

Trigonometric Ratios of Standard Angles Class 9 ICSE ML Aggarwal img 38

Answer :

∆ABC and ∆ABD are right angled triangles in which ∠A = 90° and ∠B = 90°

BC = 2√3 cm. AC and BD intersect each other at E at right angle and ∠CAB = 30°.

Now in right ∆ABC, we have

tan θ = BC/AB

⇒ tan 30° = 2√3/ AB

⇒ 1/√3 = 2√3/ AB

⇒ AB = 2√3 × √3 = 2 × 3 = 6 cm.

In ∆ABE, ∠EAB = 30° and ∠EAB = 90°

∠ABE or ∠ABD = 180° – 90° – 30°

= 60°
Now in right ∆ABD, we have
tan 60° = AD/AB

⇒ √3 = AD/6

Thus, AD = 6√3 cm.

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