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Multiple choice maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

The angle subtended at the centre of a circle of radius $3cm$ by an arc of length $1cm$ is:

  1. $\cfrac { { 30 }^{ o } }{ \pi } $
  2. $\cfrac { { 60 }^{ o } }{ \pi } $
  3. ${ 60 }^{ o }$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Angle subtended at the centre  of circle is $\theta =\dfrac { l }{ r }$ 

$\Rightarrow \theta =\dfrac { 1 }{ 3 }$  
Now, $\pi$ radian $ =180^{o}$ 
$\Rightarrow \frac { 1 }{ 3 }$ radian $=180^{o}\times \dfrac { 1 }{ 3\pi  } =\dfrac { 60^{o} }{ \pi  }$ 
Hence, option B is correct.

Multiple choice maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

Write True or False:

The tangent to the circumcircle of an isosceles $\triangle ABC$ at A, in which $AB = AC$, is parallel to BC.

  1. True

  2. False

  3. Ambiguous

  4. Data insufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given-

PQ is a tangent to a circle at a point A when the circle is a circumcircle of the isosceles $\Delta ABC$.
$ AB=AC.$ 
To find out -
The statement, $PQ\parallel BC$, is true or not.
Justification-
In $\Delta ABC$ we have,
$ AB=AC.$ 
$\therefore  \angle ABC=\angle ACB$    ...(base angles of an isosceles triangle)    ........(i)
Again, PQ is the tangent to the circle at A & AB is a chord drawn from A. 
And AB subtends \angle ACB to the corresponding alternate segment of the circle.
$ \therefore  \angle PAB=$ corresponding alt. segment $\angle ACB$ ......(ii)
So, from (i) & (ii), 
$\angle PAB=\angle ABC.$ 
But they are alternate angles $\Longrightarrow  PQ\parallel BC$ 
$\therefore$ The statement, $PQ\parallel BC$, is true.