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 - Geometry Mathematics 2

Advertisements
Advertisements
Sum

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

Advertisements

Solution

Proof: In given figure,

`{:("AF" = "AE"),("FB" = "BD"),("EC" = "DC"):}}`  .....(i) [Tangent Segment theorem]

In ▢ODCE,

∠ECD = 90°    ......[∠ACB = 90°, A–E–C, B-D–C]

`{:(∠"ODC" = 90^circ),(∠"OEC" = 90^circ):}}`  ......[Tangent theorem]

∴ ∠EOD = 90° …[Ramining angle of ▢ODCE]

∴ ▢ODCE is a rectangle.

. Also, OE = OD = r  ......[Radii of the same circle]

∴ ▢ODCE is a square     ......`[("A Rectangle is square if it's"),("adjcent sides are congruent")]`

∴ OE = OD = CD = CE = r    ......(ii) [sides of the square]

Consider R.H.S. = a + b – c

= BC + AC – AB

= (BD + DC) + (AE + EC) – (AF + FB) ......[B–D–C, A–E–C, A–F–B]

= (FB + r) + (AF + r) – (AF + FB)   ......[From (i) and (ii)]

= FB + r + AF + r – AF - FB

= 2r

= L.H.S

∴ 2r = a + b – c

  Is there an error in this question or solution?
Chapter 3: Circle - Q.7

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 Fig. 8, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.


Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y. Hence find the radius of the circle.


In the given figure, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects PQ at A and RS at B. Prove that ∠AOB = 90º


In fig. XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that, XA + AR = XB + BR.


Fill in the blank

The angle between tangent at a point on a circle and the radius through the point is ........


O is the center of a circle of radius 8cm. The tangent at a point A on the circle cuts a line through O at B such that AB = 15 cm. Find OB


If PT is a tangent at T to a circle whose center is O and OP = 17 cm, OT = 8 cm. Find the length of tangent segment PT.


In the fig. ABC is right triangle right angled at B such that BC = 6cm and AB = 8cm. Find the radius of its in circle.


If ΔABC is isosceles with AB = AC and C (0, 2) is the in circle of the ΔABC touching BC at L, prove that L, bisects BC.


ture or false v

The degree measure of a semi-circle is 180°.


Two parallel chords are drawn in a circle of diameter 30.0 cm. The length of one chord is 24.0 cm and the distance between the two chords is 21.0 cm; find the length of another chord.


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 given figure ABC is an isosceles triangle and O is the centre of its circumcircle. Prove that AP bisects angle BPC .


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°


In the given figure, a circle with center O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29cm, AD = 23cm, ∠B = 90° and DS=5cm then find the radius of the circle.

 


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


In Fig. 4, a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then find the lengths of AD, BE and CF.


The circumference of a circle is 22 cm. The area of its quadrant (in cm2) is

 


In Figure 3, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of the side AD.


In the given figure, ΔPQR is an isosceles triangle with PQ = PR and m ∠PQR = 35°. Find m ∠QSR and QTR.


In the given figure, AB and CD are diameters of a circle with centre O. If ∠OBD = 50°, find ∠AOC.


The greatest chord of a circle is called its


In the given figure, if chords AB and CD of the circle intersect each other at right angles, then x + y =


In the given figure, chords AD and BC intersect each other at right angles at a point P. If ∠DAB = 35°, then 


A triangle PQR is drawn to circumscribe a circle of radius 8 cm such that the segments 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 ∆PQR is 336 cm2, find the sides PQ and PR.


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 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 area of a circle of radius 7 cm.


In the given figure, chord EF || chord GH. Prove that, chord EG ≅ chord FH. Fill in the blanks and write the proof. 


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. 


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.


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 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.


Use the figure given below to fill in the blank:

EF is a ______ 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.


Draw circle with diameter:  8.4 cm

In above case, measure the length of the radius of the circle drawn.


Draw a circle of radius 6 cm. In the circle, draw a chord AB = 6 cm.

(i) If O is the center of the circle, join OA and OB.

(ii) Assign a special name to ∆AOB

(iii) Write the measure of angle AOB.


Draw a circle of radius 3.6 cm. In the circle, draw a chord AB = 5 cm. Now shade the minor segment 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.


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:

The longest chord of a circle is its diameter.


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 ______________


Find the diameter of the circle

Radius = 10 cm


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


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?


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, 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.


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.


In the given figure, if ZRPS = 25°, the value of ZROS is ______ 

 


If a number of circles pass through the endpoints P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of PQ.


If AB is a chord of a circle with centre O, AOC is a diameter and AT is the tangent at A as shown in figure. Prove that ∠BAT = ∠ACB


If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is ______.


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


If AOB is a diameter of a circle and C is a point on the circle, then AC2 + BC2 = AB2.


If A, B, C and D are four points such that ∠BAC = 45° and ∠BDC = 45°, then A, B, C, D are concyclic.


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.


From the figure, identify the centre of the circle.

 


From the figure, identify a diameter.

 


From the figure, identify a point in the exterior.


From the figure, identify a sector.


From the figure, identify a segment.


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:

Two diameters of a circle will necessarily intersect.


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.

The circumcentre of a triangle is the point which is ______.


Share
Notifications



      Forgot password?
Use app×