The common point of the tangent and the circle is called

1

(a)

point of contact.

The length of the tangents drawn from an external point to a circle are _____________

1

(a)

equal

If a line and a circle have no point common, then the line lies ______________

1

(a)

outside the circle.

At the point of contact, angle between the tangent and the radius is is ___________

1

(a)

A right angle \(\left( { 90 }^{ \circ } \right) \).

In the figure, QR is a common tangent to given circle which meet at T. Tangent at T meets QR at P. If QP = 3.8 cm, then find length of QR.

1

(a)

QP = 3.8
QP = PT
(Length of tangents from external points are equal)
\(\Rightarrow\)  PT = 3.8 cm
PR = PT = 3.8 cm
\(\Rightarrow\)  QR = 7.6 cm

The distance between two parallel tangents drawn to a circle is equal to the diameter of the circle.

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(a) False

(b) True

A tangent line intersects the circle in two points.

1

(a) True

(b) False

The parallelogram circumscribing a circle is a rectangle.

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(a) True

(b) False

The length of the tangent is the length of the segment from an external point to the point of contact.

1

(a) False

(b) True

 A circle can have maximum two tangents.

1

(a) True

(b) False

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

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(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 

From an external point P, k tangents can be drawn to a circle. Find the value of k.

1

(a)

From an external point P only two tangents can be drawn to a  circle.
Value of k = 2.

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 given figure, O is the centre of the circle, AB is a chord and AT is the tangent at If \(\angle AOB=100^0\) then find \(\angle BAT\)

1

(a)

\(\angle \)AOB = 100°
\(\angle \)AOB + \(\angle \)OAB + \(\angle \)OBA = 180°
⇒ 100° + x + x = 180° [Let \(\angle \)OAB = x = \(\angle \)OBA]
angle opp. to equal side
⇒ 2x = 180° - 100°
⇒ ​​​​​​​2x = 80° ⇒ x = 40°
\(\angle \)OAB + \(\angle \)BAT = 90°​​​​​​​
⇒ ​​​​​\({ 40 }^{ \underset { \_ }{ O } }\) + BAT = 90°
⇒ \(\angle \)BAT = 50 °​​​​

In figure PQ and PR are tangents to circle with centre A.If \(\angle QPA=27^0,\)then find \(\angle QAR.\)

1

(a)

\(\angle \)QPA = \(\angle \)RPA 

⇒ \(\angle \)RPA = 27°
\(\angle \)QPR = \(\angle \)QPA + RPA 
= 27° + 27° = 54°
Now,  \(\angle \)QAR + \(\angle \)QPR = 180°
⇒ \(\angle \)QAR = 180° - 54° = 126°

Two parallel lines touch the circle at points A and B respectively. If area of the circle is \(25\pi cm^2\) find AB.

1

(a)

Let radius of circle = R
∴  \(\pi\)R2 = 25\(\pi\)
⇒ R = 5 cm
Distance between parallel tangents 
= 2 x 10 cm. 

Prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

4

(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.

4

(a)

AB is a diameter of a circle. AH and BK are perpendicular from A and B respectively to the tangent at P.Prove that AH + BK = AB.

4

(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)

In the given figure AB is a chord of a circle, with centre O, such that AB = 16 cm and radius of circle is 10 cm. Tangents at A and B intersect each other at P. Find the length of PA.

5

(a)

a, b and c are the sides of a right triangle, where c is the hypotenuse. A circle, of radius r, touches the sides of the triangle. Prove that \(r=\frac { a+b-c }{ 2 } \)

5

(a)