In Fig. There Are Two Concentric Circles with Centre O of Radii 5cm and 3cm. from an External Point P, Tangents Pa and Pb Are Drawn to These Circles If Ap = 12cm, Find the Tangent Length of Bp. - Mathematics

Advertisements
Advertisements

In fig. there are two concentric circles with Centre O of radii 5cm and 3cm. From an
external point P, tangents PA and PB are drawn to these circles if AP = 12cm, find the
tangent length of BP.

Advertisements

Solution

OA = 5 cm

OB = 3 cm

AP = 12 cm

BP = ?

We know that

At the point of contact, radius is perpendicular to tangent.

For circle 1, ΔOAP is right triangle

By Pythagoras theorem, 𝑂𝑃2 = 𝑂𝐴2 + 𝐴𝑃2

⇒ 𝑂𝑃2 = 52 + 122 = 25 + 144

= 169

⇒ OP = `sqrt(169)` = 13 𝑐𝑚

For circle 2, ΔOBP is right triangle by Pythagoras theorem,

𝑂𝑃2 = 𝑂𝐵2 + 𝐵𝑃2

132 = 32 + 𝐵𝑃2

𝐵𝑃2 = 169 − 9 = 160

𝐵𝑃 = `sqrt(160) = 4sqrt(10)` 𝑐𝑚

  Is there an error in this question or solution?
Chapter 8: Circles - Exercise 8.2 [Page 35]

APPEARS IN

RD Sharma Class 10 Maths
Chapter 8 Circles
Exercise 8.2 | Q 26 | Page 35

RELATED QUESTIONS

In Figure 1, common tangents AB and CD to the two circles with centres 01and 0intersect at E. Prove that AB = CD.


In the given figure, the incircle of ∆ABC touches the sides BC, CA and AB at D, E, F respectively. Prove that AF + BD + CE = AE + CD + BF = `\frac { 1 }{ 2 } ("perimeter of ∆ABC")`


PA and PB are tangents from P to the circle with centre O. At point M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.


Write True or False. Give reasons for your answers.

If a circle is divided into three equal arcs, each is a major arc.


Write True or False. Give reasons for your answers.

A chord of a circle, which is twice as long as its radius, is a diameter of the circle.


Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at center.


From an external point P, tangents PA and PB are drawn to the circle with centre O. If CD is the tangent to the circle at point E and PA = 14 cm. Find the perimeter of ABCD.


In fig. a circle touches all the four sides of quadrilateral ABCD with AB = 6cm, BC = 7cm, CD = 4cm. Find AD.


Suppose You Are Given a Circle. Give a Construction to Find Its Centre.


In the following figure, OABC is a square. A circle is drawn with O as centre which meets OC
at P and OA at Q. Prove that:
(i) ΔOPA ≅ ΔOQC, (ii) ΔBPC ≅ ΔBQA.


In the following figure, AB is the diameter of a circle with centre O and CD is the chord with length equal to radius OA.

Is AC produced and BD produced meet at point P; show that ∠APB = 60°


Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.


In fig. 3 are two concentric circles of radii 6 cm and 4 cm with centre O. If AP is a tangent to the larger circle and BP to the smaller circle and length of AP is 8 cm, find the length of BP ?


A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal ?


If the difference between the circumference and the radius of a circle is 37 cm, then using`22/7`, the circumference (in cm) of the circle is:


In Fig. 4, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.


In the given figure, ABCD is a cyclic quadrilateral. If ∠BCD = 100° and ∠ABD = 70°, find ∠ADB.


In the given figure, O is the centre of the circle. If ∠CEA = 30°, Find the values of xy and z.

 


A chord of length 14 cm is at  a distance of 6 cm from the centre of a circle. The length of another chord at a distance of 2 cm from the centre of the circle is


In the given figure, ABC is a right triangle right-angled at B such that BC = 6 cm and AB = 8 cm. Find the radius of its incircle.


In Fig. 8.78, there are two concentric circles with centre O. PRT and PQS are tangents to the inner circle from a point P lying on the outer circle. If PR = 5 cm, find the length of PS.


Choose correct alternative answer and fill in the blank. 

Radius of a circle is 10 cm and distance of a chord from the centre is 6 cm. Hence the length of the chord is .........


Radius of a circle is 10 cm and distance of a chord from the centre is 6 cm. Hence the length of the chord is ______.


The point of concurrence of all angle bisectors of a triangle is called the ______.


The circle which passes through all the vertices of a triangle is called ______.


Length of a chord of a circle is 24 cm. If distance of the chord from the centre is 5 cm, then the radius of that circle is ______.


The length of the longest chord of the circle with radius 2.9 cm is ______.


Radius of a circle with centre O is 4 cm. If l(OP) = 4.2 cm, say where point P will lie.


The lengths of parallel chords which are on opposite sides of the centre of a circle are 6 cm and 8 cm. If radius of the circle is 5 cm, then the distance between these chords is ______.


Find the length of the chord of a circle in the following when: 

Radius is 13 cm and the distance from the centre is 12 cm 


AB and CD are two equal chords of a drde intersecting at Pas shown in fig. P is joined to O , the centre of the cirde. Prove that OP bisects  ∠ CPB. 


Find the diameter of the circle if the length of a chord is 3.2 cm and itd distance from the centre is 1.2 cm.


Find the area of a circle of radius 7 cm.



In the above figure, `square`XLMT is a rectangle. LM = 21 cm, XL = 10.5 cm. Diameter of the smaller semicircle is half the diameter of the larger semicircle. Find the area of non-shaded region.


In the given figure, O is the centre of a circle, chord PQ ≅ chord RS If ∠ POR = 70° and (arc RS) = 80°, find –
(1) m(arc PR)
(2) m(arc QS)
(3) m(arc QSR)  


In the given figure, seg MN is a chord of a circle with centre O. MN = 25, L is a point on chord MN such that ML = 9 and d(O,L) = 5. Find the radius of the circle. 


The figure given below shows a circle with center O in which diameter AB bisects the chord CD at point E. If CE = ED = 8 cm and EB = 4 cm,
find the radius of the circle.


In the following figure, OABC is a square. A circle is drawn with O as centre which meets OC at P and OA at Q.
Prove that:
( i ) ΔOPA ≅ ΔOQC 
( ii ) ΔBPC ≅ ΔBQA


Draw two circles of different radii. How many points these circles can have in common? What is the maximum number of common points?


Suppose you are given a circle. Describe a method by which you can find the center of this circle.


In the above figure, seg AB is a diameter of a circle with centre P. C is any point on the circle.  seg CE ⊥ seg AB. Prove that CE is the geometric mean of AE and EB. Write the proof with the help of the following steps:
a. Draw ray CE. It intersects the circle at D.
b. Show that CE = ED.
c. Write the result using the theorem of the intersection of chords inside a circle. d. Using CE = ED, complete the proof. 


In an equilateral triangle, prove that the centroid and center of the circum-circle (circumcentre) coincide.


Two concentric circles with center O have A, B, C, D as the points of intersection with the lines L shown in the figure. If AD = 12 cm and BC s = 8 cm, find the lengths of AB, CD, AC and BD.


In the given circle with diameter AB, find the value of x.


In the given figure, the area enclosed between the two concentric circles is 770 cm2. If the radius of the outer circle is 21 cm, calculate the radius of the inner circle.


ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm (not drawn to scale). Three circles are drawn touching each other with the vertices as their centres. Find the radii of the three circles.


Use the figure given below to fill in the blank:

R is the _______ of the circle.


Use the figure given below to fill in the blank:

________ is a radius of the circle.


Draw a line AB = 8.4 cm. Now draw a circle with AB as diameter. Mark a point C on the circumference of the circle. Measure angle ACB.


Construct a triangle ABC with AB = 4.2 cm, BC = 6 cm and AC = 5cm. Construct the circumcircle of the triangle drawn.


Construct a triangle PQR with QR = 5.5 cm, ∠Q = 60° and angle R = 45°. Construct the circumcircle cif the triangle PQR.


Construct a triangle ABC with AB = 5 cm, ∠B = 60° and BC = 6. 4 cm. Draw the incircle of the triangle ABC.


The center of a circle is at point O and its radius is 8 cm. State the position of a point P (point P may lie inside the circle, on the circumference of the circle, or outside the circle), when:

(a) OP = 10.6 cm

(b) OP = 6.8 cm

(c) OP = 8 cm


State, if the following statement is true or false:

If the end points A and B of the line segment lie on the circumference of a circle, AB is a diameter.


State, if the following statement is true or false:

Every diameter bisects a circle and each part of the circle so obtained is a semi-circle.


State, if the following statement is true or false:

The diameters of a circle always pass through the same point in the circle.


If the radius of a circle is 5 cm, what will its diameter be?


Draw circle with the radii given below.

2 cm


Draw circle with the radii given below.

3 cm


Draw a circle with the radii given below.

4 cm


Draw a circle of any radius. Show one diameter, one radius, and one chord on that circle.


In the table below, write the names of the points in the interior and exterior of the circle and those on the circle.

Diagram Points in the interior of the circle Points in the exterior of the circle Points on the circle
     

The diameter of the circle is 52 cm and the length of one of its chord is 20 cm. Find the distance of the chord from the centre


The chord of length 30 cm is drawn at the distance of 8 cm from the centre of the circle. Find the radius of the circle


Find the length of the chord AC where AB and CD are the two diameters perpendicular to each other of a circle with radius `4sqrt(2)` cm and also find ∠OAC and ∠OCA


A chord is 12 cm away from the centre of the circle of radius 15 cm. Find the length of the chord


In a circle, AB and CD are two parallel chords with centre O and radius 10 cm such that AB = 16 cm and CD = 12 cm determine the distance between the two chords?


Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord


A chord is at a distance of 15 cm from the centre of the circle of radius 25 cm. The length of the chord is


In the figure, O is the centre of a circle and diameter AB bisects the chord CD at a point E such that CE = ED = 8 cm and EB = 4 cm. The radius of the circle is


AD is a diameter of a circle and AB is a chord If AD = 30 cm and AB = 24 cm then the distance of AB from the centre of the circle is


The ratio between the circumference and diameter of any circle is _______


A line segment which joins any two points on a circle is a ___________


The longest chord of a circle is __________


The radius of a circle of diameter 24 cm is _______


A part of circumference of a circle is called as _______


Find the missing values in the following table for the circles with radius (r), diameter (d) and Circumference (C).

radius (r) diameter (d) Circumference (C)
15 cm    

Find the missing values in the following table for the circles with radius (r), diameter (d) and Circumference (C).

radius (r) diameter (d) Circumference (C)
    1760 cm

Find the missing values in the following table for the circles with radius (r), diameter (d) and Circumference (C).

radius (r) diameter (d) Circumference (C)
  24 m  

All the radii of a circle are _______________


The ______________ is the longest chord of a circle


A line segment joining any point on the circle to its center is called the _____________ of the circle


A line segment with its end points on the circle is called a ______________


Twice the radius is ________________


Find the diameter of the circle

Radius = 8 cm


Find the diameter of the circle

Radius = 6 cm


Find the radius of the circle

Diameter = 24 cm


Find the radius of the circle

Diameter = 30 cm


Find the radius of the circle

Diameter = 76 cm


A, B, C are any points on the circle with centre O. If m(arc BC) = 110° and m(arc AB) = 125°, find measure arc AC.


In the adjoining figure, seg DE is the chord of the circle with center C. seg CF⊥ seg DE and DE = 16 cm, then find the length of DF?


In figure, chords AC and DE intersect at B. If ∠ABE = 108°, m(arc AE) = 95°, find m(arc DC).


In figure, O is the centre of a circle, chord PQ ≅ chord RS. If ∠POR = 70° and (arc RS) = 80°, find

(i) m(arc PR)

(ii) m(arc QS) 

(iii) m(arc QSR)


In the figure, segment PQ is the diameter of the circle with center O. The tangent to the tangent circle drawn from point C on it, intersects the tangents drawn from points P and Q at points A and B respectively, prove that ∠AOB = 90°


Given: A circle inscribed in a right angled ΔABC. If ∠ACB = 90° and the radius of the circle is r.

To prove: 2r = a + b – c


In a circle with centre P, chord AB is parallel to a tangent and intersects the radius drawn from the point of contact to its midpoint. If AB = `16sqrt(3)`, then find the radius of the circle


In the figure, O is the center of the circle. Line AQ is a tangent. If OP = 3, m(arc PM) = 120°, then find the length of AP.


In the figure, O is the centre of the circle, and ∠AOB = 90°, ∠ABC = 30°. Then find ∠CAB.


In the figure, a circle touches all the sides of quadrilateral ABCD from the inside. The center of the circle is O. If AD⊥ DC and BC = 38, QB = 27, DC = 25, then find the radius of the circle.


Circles with centres A, B and C touch each other externally. If AB = 36, BC = 32, CA = 30, then find the radii of each circle.


C(O, r1) and C(O, r2) are two concentric circles with r1 > r2 AB is a chord of C(O, r1) touching C(O, r2) at C then ______


If the angle between two radii of a circle is 130°, then the angle between the tangents at the ends of the radii is ______


If d1, d2 (d2 > d1) be the diameters of two concentric circles and c be the length of a chord of a circle which is tangent to the other circle, then ______ 


In the adjoining figure ‘O’ is the center of the circle, ∠CAO = 25° and ∠CBO = 35°. What is the value of ∠AOB?  


The length of the tangent from point A to a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is ______  


AD is a diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, the distance of AB from the centre of the circle is ______.


In figure, ∠AOB = 90º and ∠ABC = 30º, then ∠CAO is equal to ______.


On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that ∠BAC = ∠BDC.


The circumcentre of the triangle ABC is O. Prove that ∠OBC + ∠BAC = 90º.


In the given figure, O is the centre of the circle. Name all chords of the circle.


In the given figure, O is the centre of the circle. Name a chord, which is not the diameter of the circle.


From the figure, identify the centre of the circle.

 


From the figure, identify three radii.

 


From the figure, identify a diameter.

 


From the figure, identify a chord.


From the figure, identify two points in the interior.


From the figure, identify a point in the exterior.


From the figure, identify a segment.


Say true or false:

Two diameters of a circle will necessarily intersect.


A circle of radius 3 cm with centre O and a point L outside the circle is drawn, such that OL = 7 cm. From the point L, construct a pair of tangents to the circle. Justify LM and LN are the two tangents.


If radius of a circle is 5 cm, then find the length of longest chord of a circle.


Assertion (A): If the circumference of a circle is 176 cm, then its radius is 28 cm.

Reason (R): Circumference = 2π × radius of a circle.


Share
Notifications



      Forgot password?
Use app×