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. - Mathematics

Advertisements
Advertisements
Short Note

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.

Advertisements

Solution

From the property of tangents we know that the length of two tangents drawn to a circle from the same external point will be equal. Therefore, we have

BQ = BP

Let us denote BP and BQ by x

AP = AR

Let us denote AP and AR by y

RC = QC

Let us denote RC and RQ by z

 

We have been given that ΔABC  is a right triangle and BC = 6 cm and AB = 8 cm. let us find out AC using Pythagoras theorem. We have,

`AC^2=AB^2+BC^2`

`AC^2=6^2+8^2`

`AC^2=36+64`

`AC^2=100`

`AC= sqrt100`

`AC=10`

Consider the perimeter of the given triangle. We have,

AB + BC + AC = 8 + 6 + 10

AB + BC + AC = 24

Looking at the figure, we can rewrite it as,

AP + PB + BQ + QC + AR + RC = 24

Let us replace the sides with the respective x, y and z which we have decided to use.

`y+x+x+z+y+z=24`

`2x+2y+2z=24`

`2(x+y+z)=24`

`x+y+z=12`

Now, consider the side AC of the triangle.

AC = 10

Looking at the figure we can say,

AR + RC = 10

y + z = 10 …… (2)

Now let us subtract equation (2) from equation (1). We have,

x + y + z = 12

y + z = 10

After subtracting we get,

x = 2

That is,

BQ = 2, and

BP = 2

Now consider the quadrilateral BPOQ. We have,

BP = BQ (since length of two tangents drawn to a circle from the same external point are equal)

Also,

PO = OQ (radii of the same circle)

It is given that `∠PBQ= 90^o` 

From the property of tangents, we know that the tangent will be at right angle to the radius of the circle at the point of contact. Therefore,

`∠OPB= 90^o` 

`∠OQB= 90^o` 

We know that sum of all angles of a quadrilateral will be equal to `360^o`. Therefore,

`∠PBQ+∠OPB+∠OQB+∠POQ=360^o`

`90^o + 90^o +90^o + ∠POQ= 360^o`

`270^o + ∠POQ = 360^o`

`∠ POQ= 90^o`

Since all the angles of the quadrilateral are equal to `90^o`and the adjacent sides also equal, this quadrilateral is a square. Therefore, all sides will be equal. We have found out that,

BP = 2 cm

Therefore, the radii

PO = 2 cm

Thus the radius of the incircle of the triangle is 2 cm.

  Is there an error in this question or solution?
Chapter 8: Circles - Exercise 8.2 [Page 34]

APPEARS IN

RD Sharma Class 10 Maths
Chapter 8 Circles
Exercise 8.2 | Q 15 | Page 34

RELATED QUESTIONS

In Fig. 2, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ.
 


In the given figure, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS.


Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.


Prove that there is one and only one tangent at any point on the circumference of a circle.


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.


Fill in the blanks:

The centre of a circle lies in ____________ of the circle. 


Fill in the blanks:

The longest chord of a circle is a __________ of the circle.


Write True or False. Give reason for your answer.
A circle is a plane figure.


A chord PQ of a circle of radius 10 cm substends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle.


If AB, AC, PQ are tangents in Fig. and AB = 5cm find the perimeter of ΔAPQ.


In figure PA and PB are tangents from an external point P to the circle with centre O. LN touches the circle at M. Prove that PL + LM = PN + MN


In the given figure, if arc AB = arc CD, then prove that the quadrilateral ABCD is an isosceles– trapezium (O is the centre of the circle).


In the given figure, PA and PB are the tangent segemtns to a circle with centre O. Show that he points A, O, B and P are concyclic.


In the given figure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If ∠PBT = 30° , prove that BA : AT = 2 : 1.

 


In the given figure, PA and PB are two tangents to the circle with centre O. If ∠APB = 60°  then find the measure of ∠OAB.


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 ?


In Fig. 1, the sides AB, BC and CA of a triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11 cm, then the length of BC (in cm) is ?


A chord of a circle of radius 14 cm subtends an angle of 120° at the centre. Find the area of the corresponding minor segment of the circle. `[User pi22/7 and sqrt3=1.73]`


On a semi-circle with AB as diameter, a point C is taken, so that m (∠CAB) = 30°. Find m(∠ACB) and m (∠ABC).


An equilateral triangle ABC is inscribed in a circle with centre O. The measures of ∠BOCis


In the given figure, 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 


If \[d_1 , d_2 ( d_2 > d_1 )\] be the diameters of two concentric circle s and c be the length of a chord of a circle which is tangent to the other circle , prove that\[{d_2}^2 = c^2 + {d_1}^2\].


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 


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

Radius is 1. 7cm and the distance from the centre is 1.5 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.


If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.


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.


ABC is a right triangle in which ∠B = 90°.  If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle.


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 Fig., chords AB and CD of the circle intersect at O. AO = 5 cm, BO = 3 cm and CO = 2.5 cm. Determine the length of DO.


If O is the centre of the circle, find the value of x in each of the following figures


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:

Diameter of a circle is ______.


Use the figure given below to fill in the blank:

EF is a ______ of the circle.


Use the figure given below to fill in the blank:

________ is a radius of the circle.


Draw a circle of radius of 4.2 cm. Mark its center as O. Takes a point A on the circumference of the circle. Join AO and extend it till it meets point B on the circumference of the circle,

(i) Measure the length of AB.

(ii) Assign a special name to AB.


Mark two points A and B ,4cm a part, Draw a circle passing through B and with A as a center


Construct a triangle XYZ in which XY = YZ= 4.5 cm and ZX = 5.4 cm. Draw the circumcircle of the triangle and measure its circumradius.


The diameter of a circle is 12.6 cm. State, the length of its radius.


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 = 10 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


Circles with centres A, B and C touch each other externally. If AB = 3 cm, BC = 3 cm, CA = 4 cm, then find the radii of each circle.


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?


If O is the center of the circle in the figure alongside, then complete the table from the given information.

The type of arc

Type of circular arc Name of circular arc Measure of circular arc
Minor arc    
Major arc    

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)


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 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 with center P touches the semicircle at points Q and C having center O. If diameter AB = 10, AC = 6, then find the radius x of the smaller circle.


A point P is 13 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 12cm. Find the radius of the circle.


In the given figure, AB is the diameter of the circle. Find the value of ∠ACD.


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, if ∠DAB = 60º, ∠ABD = 50º, then ∠ACB is equal to ______.


A circle of radius 3 cm can be drawn through two points A, B such that AB = 6 cm.


A quadrilateral ABCD is inscribed in a circle such that AB is a diameter and ∠ADC = 130º. Find ∠BAC.


In figure,O is the centre of the circle, ∠BCO = 30°. Find x and y.


Draw two acute angles and one obtuse angle without using a protractor. Estimate the measures of the angles. Measure them with the help of a protractor and see how much accurate is your estimate


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. Shade the smaller segment of the circle formed by CP.


From the figure, identify the centre of the circle.

 


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 sector.


Draw any circle and mark

  1. it's centre
  2. a radius
  3. a diameter
  4. a sector
  5. a segment
  6. a point in its interior
  7. a point in its exterior
  8. an arc

Say true or false:

The centre of a circle is always in its interior.


A figure is in the form of rectangle PQRS having a semi-circle on side QR as shown in the figure. Determine the area of the plot.


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.


AB is a chord of a circle with centre O. AOC is diameter of circle, AT is a tangent at A.

Write answers to the following questions:

  1. Draw the figure using the given information.
  2. Find the measures of ∠CAT and ∠ABC with reasons.
  3. Whether ∠CAT and ∠ABC are congruent? Justify your answer.

Share
Notifications



      Forgot password?
Use app×