Question

In Fig. AP is a tangent to the circle with centre O such that OP = 4 cm and ∠OPA = 30°. Then, AP =

A. 2√2 cm 

B. 2 cm 

C. 2√3cm 

D. 3√2 cm

Answer 1

Answer is C. 2√3cm

Given: 

OP = 4 cm 

∠OPA = 30°

Property: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency. 

By above property, ∆POA is right-angled at ∠OAP (i.e., ∠OAP = 90°). 

Now we know that

cos θ = \(\frac{Base}{Hypotnuse}\)

Therefore,

Cos ∠P = \(\frac{AP}{OP}\)

⇒ Cos 30° = \(\frac{AP}{4}\)

⇒ \(\frac{\sqrt{3}}{2}\) = \(\frac{AP}{4}\)

⇒ AP = \(\frac{4\times\sqrt{3}}{2}\)

⇒ AP = 2√3 cm 

Hence, AP = 2√3 cm

Answer 2

in  triangle OPA, ∠OPA = 30°, ∠OAP = 90°

let AP=x cm

then cos(∠OPA )=b/h=AP/OP

=>cos(30⁰)=x/4

=>x=4 * (√3/2)=2√3    (ans)