Tag: circle and its elements

Questions Related to circle and its elements

Write True or False:

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

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

  1. True

  2. False

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: A
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.

In $\bigodot (P, 6)$, the length of an arc is $\pi$. Then the arc subtends an angle of measure ___at the center.

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

  1. 30

  2. 60

  3. 90

  4. 120

Show answer
Correct Option: A
Explanation:

Given : $r=6$

$l=\pi $
We know that $l=r\theta $
$\Rightarrow \cfrac { \pi  }{ 6 } =\theta $
$\theta ={ 30 }^{ 0 }$
$\therefore$ The arc subtends ${ 30 }^{ 0 }$ at $P$.