Question

A vertical pole stands at a point A on the boundary of circular park of radius a and subtends and angle beta at another point B on the boundary.IF the chord AB subtends an angle alpha at the centre of park ,th height of the pole is A)2asin(alpha/2)tan(beta) B)2acos(alpha/2)tan(beta )​

Answer 1

Answer:

Height of the pole is option A. $2a.sin(\frac{\alpha}{2}).tan(\beta)$

Step-by-step explanation:

A vertical pole TA is standing at point A which subtends ∠TBA = β.

Chord AB subtends an angle at the center = α.

We have to find the height of the tower TA from the given picture attached.

Applying sine rule in ΔABO,

$\frac{sin\alpha}{AB}=\frac{sinx}{a}$

Since OA = OB = a [Radii of the circle]

Therefore, ∠OBA ≅ ∠OAB ≅ x

∠OAB + ∠OBA + α = 180°

x + x + α = 180°

2x + α = 180°

2x = 180° - α

x = 90° - $\frac{\alpha}{2}$

Now we place the value of x in the equation

$\frac{sin\alpha}{AB}=\frac{sin(90-\frac{\alpha}{2})}{a}$

$\frac{sin\alpha}{AB}=\frac{cos\frac{\alpha}{2} }{a}$

AB = $a[\frac{sin\alpha}{\cos(\frac{\alpha}{2})}]$

     = $\frac{2asin(\frac{\alpha}{2})cos(\frac{\alpha}{2})}{cos(\frac{\alpha}{2})}$

     = $2asin(\frac{\alpha}{2})$

Now in the right angle ΔTAB,

$tan(\beta)=\frac{TA}{AB}$

TA = AB.tan(β)

TA = $2a.sin(\frac{\alpha}{2}).tan(\beta)$

Therefore, Option A. will be the answer.      

Learn more about the chords and angles of a circle from https://brainly.in/question/8978727