In an Equilateral Triangle, Prove that the Centroid and Center of the Circum-circle (Circumcentre) Coincide. - Mathematics

Advertisements
Advertisements
Sum

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

Advertisements

Solution

Given: An equilateral triangle ABC in which D, E, and F are the midpoints of sides BC, CA and AB respectively.

To prove: The centroid and circumference are coincident.

Construction: Draw medians AD, BE and CF.

Proof:
Let G be the centroid of ΔABC i.e., the point of intersection of AD, BE, and CF. In triangles BEC and BFC, we have
∠ B = ∠ C = 60°
BC = BC
and BF = CE    ...[ ∵ AB = AC ⇒ `1/2"AB" = 1/2` AC ⇒ BF = CE ]
∴ ΔBEC = ΔBFC
⇒ BE = CF           ...(i)
Similarly,
Δ CAF and Δ CAD
⇒ CF = AD           ...(ii)

From (i) and (ii),
AD = BE = CF
⇒ `2/3"AD" = 2/3 BE = 2/3"CF"`

CG = `2/3"CF"`

GA = `2/3"AD"`

GB = `2/3 "BE"`

GA = GB = GC

⇒ G is the equidistant from the vertices
⇒ G is the circumcentre of ΔABC.
Hence, the centroid and circumcentre are coincident.

  Is there an error in this question or solution?
Chapter 15: Circles - Exercise 1

APPEARS IN

ICSE Class 10 Mathematics
Chapter 15 Circles
Exercise 1 | Q 15

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.


Fill in the blanks:

A circle divides the plane, on which it lies, in __________ parts.


Write True or False. Give reason for your answer. 

Sector is the region between the chord and its corresponding arc.


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


Two tangent segments PA and PB are drawn to a circle with center O such that ∠APB =120°. Prove that OP = 2AP


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.


Fill in the blank:

An arc is a ................ when its ends are the ends of a diameter.


true or false 

A circle is a plane figure.


Prove that two different circles cannot intersect each other at more than two points.


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


In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BC and DC into which BC is divided by the point of contact D, are of
lengths 6cm and 9cm respectively. If the area of 2 ΔABC = 54cm2 then find the lengths of sides AB and AC.


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


In the given figure, two tangents RQ, and RP and RP are drawn from an external point R to the circle with centre O. If ∠PRQ =120° , then prove that OR = PR + RQ.


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. 5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.


In the given figure, ΔABC is an equilateral triangle. Find mBEC


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


The length of three concesutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm, and 7 cm respectively. Determine the length of the fourth side.


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.


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


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


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:

Tangent to 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 chord of the circle.


Use the figure given below to fill in the blank:

If the length of RS is 5 cm, the length of PQ = _______


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.


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


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.


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 = 8 cm


Find the diameter of the circle

Radius = 6 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.


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 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 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 ______ 

 


A line through the point of contact and passing through centre of the circle is known as ______


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 the angle between two tangents drawn from a point P to a circle of radius ‘a’ and centre ‘O’ is 90°, then OP = ______ 


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 ______  


The length of tangent from an external point on a circle is always greater than the radius of the circle.


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


Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.


In figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to ______.


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


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 all radii 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.


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 three radii.

 


From the figure, identify a diameter.

 


From the figure, identify a chord.


From the figure, identify a sector.


Is every chord of a circle also a diameter?


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.


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


Share
Notifications



      Forgot password?
Use app×