The common point of the tangent and the circle is called

1

(a)

point of contact.

A line segment drawn through the end of a radius and perpendicular to it, is a ___________to the circle.

1

(a)

tangent line.

In the given figure, AOB is a diameter of the circle with centre O and AC is a tangent to the circle at A. If \(\angle BOC\) = 130°, then find \(\angle ACO\)

1

(a)

\(\angle OAC={ 90 }^{ ° }\)  (as radius \(\bot\) r tangent)
\(\angle BOC=\angle OAC+\angle ACO\)
(Exterior angle property)
\(\Rightarrow \quad { 130 }^{ ° }={ 90 }^{ ° }+\angle ACO\)
\(\Rightarrow \quad \angle ACO={ 130 }^{ ° }+{ 90 }^{ ° }\)
\({ =40 }^{ ° }\)

In the given figure, PQ and PR are tangents to the circle with centre O such that \(\angle QPR={ 50 }^{ ° }\)then find \(\angle OQR\)

1

(a)

\(\angle QPR={ 50 }^{ ° }\)          (Given)
\(\therefore \quad \angle QOR={ 180 }^{ ° }-{ 50 }^{ ° }={ 130 }^{ ° }\)
From \(\triangle OQR\)
\(\Rightarrow \quad \angle QOR=\angle ORQ=\frac { { 180 }^{ ° }-{ 130 }^{ ° } }{ 2 } \)
\(=\frac { { 50 }^{ ° } }{ 2 } ={ 25 }^{ ° }\)

 A circle can have maximum two tangents.

1

(a) True

(b) False

The common point of a tangent to a circle with circle is called point of contact.

1

(a) False

(b) True

In the figure, PT is a tangent to the circle with centre O such hat OP is 4 cm and \(\angle OPT={ 30 }^{ \circ }\) , then length of tangent is 5 cm.

1

(a) True

(b) False

The common point of a tangent and a circle is called point of contact.

1

(a) False

(b) True

What is the distance between two parallel tangents of a circle of radius 7cm?

1

(a)

parallel tangents of a circle can be drawn only at the end points of the diameter 
⇒  I₁ || I₂
⇒  Distance between I₁ and I₂ = AB = Diameter of the circle 
= 2 x r = 2 x 7 cm = 14 cm 

How many common tangents can be drawn to two circles touching internally?

1

(a)

I common tangent can be drawn as shown in the figure. 

PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that \(\angle POR=110^0\)Find \(\angle OPQ.\)

1

(a)

Given: PQ is a tangent to the circle with centre O from a point P. QCR is a diameter of the circle
and \(\angle \)POR = 110°. 
To find: \(\angle \)OPQ 

Sol. POR = 110°
QR is the diameter of the circle.
⇒ \(\angle \)1 + \(\angle \)2 = 180°  [Linear pair axiom]
⇒ \(\angle \)1 + 110° = 180° ⇒ \(\angle \)1 = 70°
\(\angle \)OQP = 90°
(Tangent makes 90° angle with the radius at the point of contact).
In \(\triangle\)OPQ
\(\angle \)1 + \(\angle \)OQP + \(\angle \)QPO = 180°      [Angle sum properety]
⇒ 70° + 90° + \(\angle \)QOP = 180°
⇒  \(\angle \)OPQ = 180° - 160°
⇒ \(\angle \)OPQ = 20°

In the figure PA and PB are tangents to the circle with centre O.If  \(\angle APB=60^0\), then find \(\angle OAB.\)

1

(a)

Given \(\angle \)APB = 60°
∵  \(\angle \)APB + \(\angle \)PAB + \(\angle \)PBA = 180°   [∵ PA = PB ⇒ \(\angle \)PAB = \(\angle \)PBA]
⇒ \(\angle \)APB + x + x = 180°
⇒ APB + 2x = 180°
⇒ 2x = 180° - 60°
⇒ 2x = 120° ⇒ x = 120°/2 = 60°
Also \(\angle \)OAP = 90°
⇒ \(\angle \)OAB + \(\angle \)PAB = 90°
\(\angle \)OAB + 60° = 90° ⇒ \(\angle \)OAB = 30°

The length of a tangent from a point A at distance 5cm from the centre of the circle is 4cm.Find the radius of the circle.

2

(a)

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

2

(a)

In the figure, a circle is inscribed in a quadrilateral ABCD in which\(\angle 90^0\).If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius(r) of the circle.

3

(a)

In the given figure, TBP and TCQ are tangents to the circle whose centre isO.Also \(\angle PBA=60^0\ and \ \angle ACQ=70^0.\)Determine \(\angle BAC\ and \ \angle BTC.\)

3

(a)

In figure, the sides AB, BC and CA of triangle ABC touch a circle with centre O and radius r at P, Q and R respectively.Prove that
(i) AB + CQ = AC + BQ
(ii)area (\(\Delta\)ABC) = \(1\over2\) (perimeter of \(\Delta\)ABC) x r

4

(a)

PQR is a right angled triangle right angled at Q.PQ=5cm, QR=12cm.A circle with centre O is inscribed in \(\Delta\)PQR, touching its ll sides.Find the radius of the circle.

4

(a)