M4maths Previous Puzzles

12 Jul 2016 Perimeter of Triangle

The area of a circle inscribed in an equilateral triangle is 346.5 cm^2. Find the perimeter of the triangle. (Use Pi = 22/7 and Sqrt3 = 1.73).

OPtion

1) 102.45 cm
2) 104.33 cm
3) 89.54 cm
4) 108.99 cm
5) 94.35 cm
6) 72.66 cm
7) 112.33 cm
8) 110.5 cm
9) 86.5 cm
10)None of these

Solution

By using figure
The incentre of the triangle is coincident with centroid (as it is an equilateral triangle)
Also I divided AD in the ratio 2:1 (I is centroid)
We have, area of circle = 346.5 cm^2
=> Pi r^2 = 346.5
r^2 = 346.5*7/22 = 110.25
r = 10.5
In triangle ABC
We have ID = AI = 1:2
ID = r = 1/3 AD
Also as triangle ABC is equilateral
AD = (Sqrt3/2)*a (From pythagoras theorem is Triangle ADB)
r = Sqrt3*a/2 * 1/3 = a/2*Sqrt3
a = 2*Sqrt3*r
Perimeter = 3a
= 3*2*Sqrt3*r = 108.99 cm.

Correct Option: 4 Best Solution (31)

Jaikumar Bhatia   8 years AGO

i) Area of inscribed circle=346.5 cm^2=Pi*r^2
=> r=10.5 cm
ii) In equilateral Triangle let h=altitude
Three altitudes bisects each other at centre of circle
& whose point of intersection divides the medians in the ratio of 2:1
r=h/3
h=3r=3*10.5=31.5 cm
iii) Let each side of Triangle be a
then,
h=root3a/2
a=2h/root3
a=2*31.5/root3
a=63/root3
a=21root3
a=36.33 cm
iv) Perimeter of equilateral triangle=3a
=>3*36.33
=>108.99 cm
Ans is 108.99 cm
option is (4)

Like? Yes | No   

Gokul Krishnan   8 years AGO

22/7*r^2=346.6cm^2
r=10.5cm
Perimeter of a triangle=3a (a=(2*h)/1.73)
where h=3r
hence ans=108.99cm

Like? Yes | No   

Razen Yadav   8 years AGO

pi*r^2=346.5
r=10.5
for equilateral triangle r=h/3(h=height)
h=31.5
again for equilateral triangle
a=2h/sqrt3 (a=side of triangle)
Perimeter of an equilateral triangle is =3a
3a=sqrt3*2*h
=108.99

Like? Yes | No   

Lakshmipathi   8 years AGO

pi r^2=346.5
r=10.5
1/rt 3=onside/another side
oneside=10.5*rt 3 (1.73)
'''''''''
108.99 cm

Like? Yes | No   

Sagar   8 years AGO

Let side be a cm of eq. Traingle
Radius of circle is a/2sqrt(3)
Area of circle = a*a*22/(7*12)

Like? Yes | No   

venkata gowtham   8 years AGO

equilateral traingle have 60degree between to sides ad you can know radius of circle and find the side length using tan formula

Like? Yes | No   

Sandip Das   8 years AGO

circle area 346.5 cm^2
pi*r^2=346.5 cm^2
r=21/2 cm
equilateral triangle height ratio is 2:1
so 2*(21/2)
height=(21/2)+21
=63/2 cm
equilateral triangle each angle is 60 degree
so tan60=(63/2)/base
base=(63/2*sqrt3)
so side=(63/2*sqrt3)+(63/2*sqrt3)
=63/sqrt3
so now perimeter is 3*side
=3*(63/sqrt3)
=sqrt3*63
=1.73*63
=108.99 cm

Like? Yes | No   

madanpal vishwakarma    8 years AGO

Given area of the circle =346.5 cm^2

So Radius 346.5 =(22/7) r^2
r=10.5
Intersection divide median in the ratio of 2:1

so height of triangle is 3r

or r=(h/3)
h=10.5 *3=31.5
Now side of triangle is a = (2*h)/sqrt3
a= (2*31.5)/sqrt3
a = 63/sqrt 3
Now perimeter of an equilateral triangle is = 3 a
= 3 *(63/sqrt 3)
= 108.99 cm
So option 4 is correct....

Like? Yes | No   

suthanth   8 years AGO

Radius of incircle of equilateral triangle of side a is a/2*sqrt(3)
So side a =36.33
Perimeter is 3a=108.99

Like? Yes | No   

NANDAN KUMAR SINGH   8 years AGO

area of circle=pi*r^2=346.5
so, r=10.5 cm
we know that in equilateral triangle each angle is 60 degree.
so,from question we can say that radius of circle is perpendicular to each side of equilateral triangle .
also radius bisect each angle .
we consider a rignt triangle OBD
in OBD
tan30=(OD/BD)
where OD=radius=10.5
BD=half distance of side BC
after calculation BD=10.5*root(3)=18.165 cm
so, BC=2*BD=2*18.165=36.33 cm

so ,perimeter of triangle=3*side length=3*BC=108.99 cm

Like? Yes | No   

Nikita Singhal   8 years AGO

area of circle = 346.5
radius of circle = 10.5 (area = pi*r^2)
altitude of equilateral triangle = 31.5 (3*radius)
side of equilateral triangle = 36.33 (2*altitude/sqrt3)
perimeter = 108.99 (3*side of triangle)

Like? Yes | No   

Ankit Kumar   8 years AGO

Area of circle=(pi)*(square of radius r) =346.5 cm sq
so,r=10.5 cm
we know that height of an equilateral triangle is divided in the ratio of 2:1 by the centre of the circle inscribed in that triangle.
Also the length of the 1st part of height(altitude) of triangle=2r and that of the 2nd part is r.
So by pythagoras theorem, length of triangle =x=2*r*(root 3)
so perimeter=3*x=108.99 cm...

Like? Yes | No   

Akshat Sood   8 years AGO

Now 22/7*r^2=3465mm

r=105mm

Now as triangle is equilateral radius is perpendicular to the side of triangle.

We know centroid divides median in ratio 2:1

BO:OE=2:1

BO=210mm

Now apply Pythagoras theorem in triangle BOF x=210?3mm or 21?3cm

As we have to find perimeter,

3x=63?3cm=108.99 cm

Option(4)

Like? Yes | No   

KOTHA VINOD KUMAR   8 years AGO

First we need find out radius of the circle from given area of the circle. Then height of the circle h=3r relation and using pygthagoars theorem u will get answer

Like? Yes | No   

Nishtha shree   8 years AGO

area of circle=346.5 cm^2
Pi*r^2=346.5=> r=10.5 cm(line from center of circle and perpendicular bisector to one side of triangle)
A line from center of circle to corner of triangle will bisect the angle
so,tan(30)=p/b=r/b=10.5 cm/b
b=18.165 cm
side of equilateral triangle=2*b
perimeter of triangle=3*(2*b)=108.99 cm

Like? Yes | No   

G.S.Kantharao   8 years AGO

Pi r^2 = 346.5cm^2
r^2= 346.5 x 7/22
= 693/2 x 7/22
= 63 x 7/4
r = 21/2
Median = 3r
Median = root 3 a/2
Root 3a = 63
Perimeter = 63 x root3
= 63 x 1.73 = 108.99

root 3 =

Like? Yes | No   

guntupalli thirumaladevi   8 years AGO

pi*r^2=346.5
r=10.5
side of the triangle(s)=2*sqrt3*r=36.33
perimeter=3s=108.99

Like? Yes | No   

milan   8 years AGO

r=10.5
tan30=r/b
one side of a triangle=2b
perimeter=3*side of a triangle

Like? Yes | No   

T.uma maheswari   8 years AGO

radius of circle=a/3.46
area of circle=(22/7)*a*a/(12)cm^2
a=36.3
perimeter of the triangle=3*a=3*36.3=108.9cm

Like? Yes | No   

nainsi gupta   8 years AGO

the inradius of equilateral triangle is=a/2root3
area of circle=pi r^2=346.5, then r=10.50
10.50=a/2root 3,then a=36.33
hence,perimeter of triangle is=3*36.33=108.99cm

Like? Yes | No   

Anish Prasad   8 years AGO

ratio of area of inscribed circle to the area of triangle is pi/3Sqrt3 so with this we can find area of the tringle which is 562.1975 cm^2
now equating this with the formula of area of an equilateral triangle ie: Sqrt3 *(side)^2/4
we would get side=21sqrt3 so perimeter will be thrice of side ie: 108.99

Like? Yes | No   

Riya Prakash Ratnaker   8 years AGO

area of circle=346.5cm^2
pi*r^2=346.5

i.e. r=10.5

In an inscribed triangle,altitude h=3*r
h=3*10.5
h=31.5

also, altitude=(root(3)/2)*side
31.5=(root(3)/2)*side

side=63/root(3)

perimeter of equilateral triangle=3*63/root(3)
=108.99cm

so answer is (4)

Like? Yes | No   

Shreeprasad Bhat   8 years AGO

since equilateral,
side = 2 * ?3 * radius
radius = ?(346.5 * 7 / 22)
side = 2 * ?3 * ?(346.5 * 7 / 22)
side = 36.33
perimeter = 3 * side
perimeter = 3 * 36.33
perimeter = 108.99

Like? Yes | No   

KUMUD UPADHYAY   8 years AGO

as in an equi triangle angle bisectors , medians and altitudes all are same so first we calculate r from pi*r^2=346.5 and we get r=10.5 cms.
Now one angle bisector makes a triangle with the centre of inscribed circle with radius as the height.
tan30�=r/x where x : half the side length of the triangle.
we get x=18.165 cms
now there 6 such x for perimeter.
perimeter=18.165*6=108.99 cms

Like? Yes | No   

arjun pandey   8 years AGO

area of triangle =22/7*r^2
r= 346.5*7/22
h=3r
h= 3*346.5*7/22
a= 2h/square root of 3
perimeter =3a
hence correct option is 4

Like? Yes | No   

sanjeet kumar   8 years AGO

Since area of circle = 346.5 cm^2.
So, Pi* r^2 = 346.5
r = 10. 5 cm
Now, tan30 = 10.5 * (side/2)
So, perimeter = 3* side = 108.99 cm

Like? Yes | No   

Rahul Das   8 years AGO

R=10.5
H=3*10.5=31.5
Side of triangle =(2*h) /sqrt3=36.41
Perimeter=3*36.41=109.1

Like? Yes | No   

Tirthankar Kundu   8 years AGO

area=346.5
pi*r^2=346.5
r=10.5
now equilateral triangle has each angle 60 degrees.
so tan 30=p/b where p=r=10.5
so we get b=10.5 * sqrt 3
each side of traingle is 2*b=21 sqrt 3 = 36.33
perimeter=3*a=3*36.33=108.99 cm

Like? Yes | No   

anil k    8 years AGO

side of trian=2sqrt3r
perimeter =3a=6sqrt3r =10.18r=10.38x10.5=108.99 cm
now r=sqrt 346.5/22x7=10.5

Like? Yes | No   

MRITUNJAY   8 years AGO

radius of circle =(346.25*7/22)^1/2
r=10.5cm

radius of incircle r= a/2*1.73
then a=r*2*1.73
a=10.5*2*1.73=36.33 cm

so perimeter of equilateral traingle =3*a=108.99 cm
option 4

Like? Yes | No (1)  

Mukund Manohar   8 years AGO

Let the radius of the circle and the side of an equilateral triangle be r and a respectively.
r = Sqrt(346.5 * 7/22)
= Sqrt(110.25)
= 10.5 cm
Now, Radius of the circle inscribed in an equilateral triangle = a/(2*Sqrt3)
=> 10.5 = a/(2*1.73)
=> a = 10.5 * 2 * 1.73
=> a = 36.33 cm
Perimeter of the triangle = 3a
= 3 * 36.33
= 108.99 cm

Like? Yes | No (1)