# Lengths of Tangents Drawn from External Point

## Trending Questions

**Q.**

O is any point inside a rectangle ABCD. Prove that OB2+OD2=OA2+OC2.

**Q.**

$3$ circles of radii $a,b,c(a<b<c)$ touch each other externally and have $X-$ axis as a common tangent then

$a,b,c$

**are in A.P**$\frac{1}{\sqrt{b}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{c}}$

$\sqrt{a},\sqrt{b},\sqrt{c}$

**are in A.P**$\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}=\frac{1}{\sqrt{a}}$

**Q.**

What is Algorithm

**Q.**

**Question 6**

If angle between two tangents drawn from a point P to a circle of radius a and centre O is 60∘ , then OP = a √3.

**Q.**

In Fig. 8.63, AB is a chord of length 16 cm of a circle of radius 10 cm. The tangents at A and B intersect at a point P. Find the length of PA.

**Q.**

In Fig. 8.61, a circle is inscribed in a quadrilateral ABCD in which ∠B=90∘. If AD =23 cm, AB =29 cm and DS =5 cm, find the radius r of the circle.

**Q.**

In the given figure, the incircle of △ABC touches the sides AB, BC and CA at the points P, Q, R respectively. Show that

AP+BQ+CR=BP+CQ+AR

=12 (Perimeter of △ABC)

[3 MARKS]

**Q.**

A circle is inscribed in ΔABC touching the sides AB, BC and CA at points D, E and F respectively. If AB= 10 cm, BC = 12 cm and CA = 8 cm, then the lengths of AD, BE and CF respectively will be

3cm, 7cm and 5cm

4cm, 6cm and 8 cm

2cm, 6cm and 7cm

5cm, 9cm and 4cm

**Q.**

From an external point P, tangents PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn, which intersects PA and PB at C and D respectively. If PA=14cm, find the perimeter of ΔPCD.

**Q.**

In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

**Q.**

In the given figure, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contant T, are of lengths 12 cm and 9 cm respectively. If the area fo \Delta PQR = 189 cm^2 then the length of side PQ is

(a) 17.5 cm (b) 20 cm (c) 22.5 cm (d) 25 cm

**Q.**

In figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.

**Q.**The area of the incircle of an equilateral triangle of side 42 cm is

(a) $22\sqrt{3}c{m}^{2}$

(b) 231 cm

^{2}

(c) 462 cm

^{2}

(d) 924 cm

^{2}

**Q.**

From an external point P, tangents PA=PB are drawn to a circle with centre O. If ∠PAB=50∘, then find ∠AOB.

**Q.**

In the adjoing figure, a circle touches all the four sides fo a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length side AD.

**Q.**

In Fig. 8.67, two tangents AB and AC are drawn to a circle with centre O such that ∠BAC=120∘. Prove that OA =2AB.

**Q.**

Quadrilateral ABCD is circumscribed to a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm then the length of AD is

(a) 3 cm (b) 4 cm (c) 6 cm (d) 7 cm

**Q.**

AB and CD are two common tangents to circles which touch each other at C. If D lies on AB such that CD = 4 cm, then find the length of AB.

4 cm

6 cm

8 cm

12 cm

**Q.**

From a point P, two tangents PA and PB are drawn to a circle with a center O. If OP= diameter of the circle, show that ΔAPB is equilateral.

**Q.**

In the given figure, PA and PB are tangents to the given circle such that PA = 5 and \angle APB = 60^o The length of chord AB is

(a) 5√2 cm (b) 5 cm

(a) 5√3 cm (a) 7.5 cm

**Q.**

In the given figure, if ∠AOD= 135∘ then ∠BOC is equal to

(a) 25∘ (b) 45∘ (c) 52.5∘ (d) 62.5∘

**Q.**Question 9

In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠AOB=90∘.

**Q.**

A triangle PQR is drawn to circumsribe a circle of radius 8 cm such that the segements QT and TR, into which QR is divided by the point of contact T, are of lengths 14 cm and 16 cm respectively. If area of ΔPQR is 336cm2, find the sides PQ and PR.

**Q.**

In the given figure, a circle touches the side DF of Δ EDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm then the permeter of Δ EDF is

(a) 9 cm (b) 12 cm (c) 13.5 cm (d) 18 cm

**Q.**

In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If ∠ TPQ = 70o, find ∠ TRQ.

**Q.**

In the given figure, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC = 4 cm then the length of AP is

(a) 15 cm (b) 10 cm (c) 9 cm (d) 7.5 cm

**Q.**

In fig., △ABC is circumscribing a circle. Find the length (in cm) of BC

**Q.**

**Question 1**

If a hexagon ABCDEF circumscribe a circle. Prove that AB + CD + EF = BC + DE + FA

**Q.**

In Fig. 8.65, BDC is a tangent to the given circle at point D such that BD=30 cm and CD=7 cm. The other tangents BE and CF are drawn respectively from B and C to the circle and meet when produced at A making BAC a right angle triangle. Calculate (i) AF (ii) radius of the circle.

**Q.**

In the given figure, a circle is inscribed in a quadrilateral ABCD touching its sides AB, BC, CD and AD at P, Q, R and S respectively. If the radius of the circle is 10 cm, BC=38 cm, PB=27 cm and AD⊥CD then the length of CD is

(a) 11 cm (b) 15 cm (c) 20 cm (d) 21 cm