Length of a Tangent

Q.

In the given figure, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP.

Q.

Two circles of radii 8 cm and 3 cm have a direct common tangent of length 10 cm. Find the distance between their centers, up to two places of decimal.

__

Q.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above, do the following:

In fig., O is the center of the two concentric circles. AB is a chord of the larger circle touching the smaller circle at C. Prove that AC = BC

Q.

Prove that the parallelogram circumscribing a circle is a rhombus.

Q.

From a point P outside a circle, with centre O, tangents PA and PB are drawn. Prove that :

(i) $\angle AOP=\angle BOP$,

(ii) OP is the $\perp$ bisector of chord AB.

Q. In the given figure, PQ is a chord of length 8 cm of a circle with centre O and radius 5 cm. If the tangents to the circle at the points P and Q intersect at 'T', then find the length of TP.

Q. If $\angle B_{1}O{A}_1={60}^{\circ }$ & radius of biggest circle is r. According to figure trapezium $A_1{B}_1{D}_1{C}_1,C_1{D}_1{D}_2{C}_2,C_2{D}_2{D}_3{C}_3.........$ and so on are obtained. Sum of areas of all the trapezium is -

1. $\frac{9r^2}{2\sqrt{3}}$
2. $\frac{9r^2}{\sqrt{3}}$
3. $\frac{r^2}{9\sqrt{3}}$
4. $\frac{r^2}{2\sqrt{3}}$

Q. Which of the following is secant to the circle given above?

1. Pc
2. CD
3. C
4. PQ

Q. In the given figure, PQ is a chord of length 8 cm of a circle with centre O and radius 5 cm. If the tangents to the circle at the points P and Q intersect at 'T', then the length of PT is

1. 20 cm
2. 40 cm
3. $\frac{40}{3}cm$
4. $\frac{20}{3}cm$

Q. A parallelogram circumscribing a circle is always a/an

1. Square
2. 
   Rhombus
3. 
   Rectangles
4. 
   Isosceles trapezium

Q.

Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above do the following:

ABC is an isosceles triangle in which AB = AC, circumscribed about a circle as shown in the fig. Prove that the base is bisected by the point of contact.

Q. Two parallel tangents of a circle meet a third tangent at points P and Q. What angle does PQ subtend at the centre?

Q. In the given figure, PQ is a chord of length 8 cm of a circle with centre O and radius 5 cm. If the tangents to the circle at the points P and Q intersect at 'T', then find the length of PT.

Q. The length of radius of a circle with centre $O$ is $5cm$. $P$ is a point at a distance of 13 from the point $O.PQ$ and $PR$ are two tangents from the point $P$ to the circles; Find the area of the quadrilaterals PQOR.

Q. In the given figure, if $PT=\frac{16}{3}cm$ and radius of the circle is $\frac{10}{3}cm$ and PLQ and QR are tangents to circle at L and R respectively, then the length of QL is

1. 6 cm
2. $5\sqrt{2}$cm
3. $6\sqrt{2}$cm
4. 5 cm

Q.

ABC is a right angled triangle, right angled at B with AB = 3 cm and BC = 4 cm. A circle which touches all the sides of the triangle is inscribed in the triangle. Calculate the radius of the circle.

__

Q.

If the sides of a quadrilateral ABCD touch a circle, prove that :

AB + CD = BC + AD.

Q.

$△$ABC is a right- angled triangle with AB = 12 cm and AC = 13 cm. A circle with center O has been inscribed inside the triangle. Calculate the radius of the inscribed circle.

__

Q.

The length of the tangent drawn to a circle with radius 7 cm from a point 25 cm away from the centre is = ___ cm.

Q. Draw a circle with radius $3cm$ and construct a pair of tangents from a point $8cm$ away from the centre.

Q.

State 'T' for true and 'F' for false.
I. If one angle of a cyclic trapezium is four times the other, then the greater one measures ${144}^{\circ }$.
II. The length of the direct common tangent to two circles with radii R and r, having distance between the centres d, is $\sqrt{d^2-(R-r{)}^2}$

1. (I) F (II)F

2. (I) T (II)F

3. (I) T (II)T

4. (I) F (II)T

Q. If figure common tangents AB and CD to two circles intersect at E. Prove that AB = CD.

Q.

PA and PB are tangents from P to the circle with center O. At point M which lies on circle, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN. [2 MARKS]

Q.

The length of the tangent drawn to a circle with radius 7 cm from a point 25 cm away from the centre is = ___ cm.

Q.

The length of the tangent drawn to a circle with radius 7 cm from a point 25 cm away from the centre is = ___ cm.

Q.

In the given figure, PQ is a chord of length 8 cm of a circle with centre O and radius 5 cm. If the tangents to the circle at the points P and Q intersect at 'T', then the length of PT is

1. 20 cm

2. $\frac{20}{3}cm$

3. $\frac{40}{3}cm$

4. 40 cm

Q.

A restaurant POQR is being constructed in the middle of a circular lake. The circle, which has its center at O is touching the mid-points of the boundary ABCD as shown in the figure. AB=CD and AD=BC. If BC=8 cm, then the area of AQOP is

1. $\frac{1}{4}\times$ Area of ABCD

2. $\frac{1}{2}\times$ Area of ABCD

3. $\frac{1}{8}\times$ Area of ABCD

4. None of these

Q.

A restaurant POQR is being constructed in the middle of a circular lake. The circle, which has its center at O is touching the mid-points of the boundary ABCD as shown in the figure. AB=CD and AD=BC. If BC=8 cm, then the area of AQOP is

1. $\frac{1}{4}\times$ Area of ABCD

2. $\frac{1}{2}\times$ Area of ABCD

3. $\frac{1}{8}\times$ Area of ABCD

4. None of these

Q.

A triangle ABC is drawn to circumscribe a circle such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 4 cm and 20 cm respectively. If the area of the triangle is 120 ${cm}^2$, what is the value of AB + AC?

1. 48 cm

2. 36 cm

3. 24 cm

4. 60 cm

Q.
AP is the tangent to the circle with centre O and radius 8 cm. If AB = 9 cm, then the length of the tangent AP is .

1. 10 cm
2. 13 cm
3. 15 cm