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Question 4 Circles - Exercise 10.4

A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed, and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

Answer:

First, draw a diagram according to the given statements. The diagram will look as follows.

Here the positions of Ankur, Syed, and David are represented as A, B, and C respectively. Since they are sitting at equal distances, the triangle ABC will form an equilateral triangle.

AD ⊥ BC is drawn. Now, AD is the median of ΔABC and it passes through the center O.

Also, O is the centroid of the ΔABC. OA is the radius of the triangle.

OA = 2/3 AD

Let the side of a triangle meters then BD = a/2 m.

Applying the Pythagoras theorem in ΔABD,

\begin{array}{l} A B^{2}=B D^{2}+A D^{2} \\ \Rightarrow A D^{2}=A B^{2}-B D^{2} \\ \Rightarrow A D^{2}=a^{2}-(a / 2)^{2} \\ \Rightarrow A D^{2}=3 a^{2} / 4 \\ \Rightarrow A D=\sqrt{3} a / 2 \\ O A=2 / 3 A D \\ \Rightarrow 20 m=2 / 3 \times \sqrt{3} a / 2 \\ \Rightarrow a=20 \sqrt{3} m \end{array}

So, the length of the string of the toy is 20√3 m.

"hello students welcome to lido q a video
session
i am seth your math tutor and question
for today is
a circular park of radius 20 meter is
situated in colony
three boys encore syed and david are
sitting at
equal distances on its boundary
each having toy telephone in his hand to
talk to each other
find the length of the string of each
phone now as per the situation we have
drawn the figure
in which angkor syria and david are
sitting at a b and c respectively
so positions of
angkor say it and
david
will be a
b and c respectively
now it is given that they are holding a
telephone with which is joined with the
wire
so toy telephone in his hand are at the
equal distances
with each each of these boys so as they
are at equal distances
triangle a b c will be equilateral
let here drop a d perpendicular to bc
where o is the center of the circle and
also the triangle
hence o is the centroid
and moreover you can see as this is a
centroid and it is perpendicular to bc
we can say that bd is equal to bc
now let one of the side of the triangle
be a
hence all the three sides of triangle
will be a a and
a meter
so you can from this you can say that bd
is equal to
a upon 2 meter now consider the right
triangle
abd in this right triangle abd
pythagoras theorem can be applied
so according to pythagoras theorem a b
square
is equal to b d square plus
a d square
here you can note one more thing that
the radius of the circular part
is 220 meter and
i can write here ao
or o a is equal to 20 meter
as oa is 20 meter we can now find
further
so oa and
we can find the relation of the centroid
so as this is a centroid we can also say
that
o a will be equal to 2 by 3 a d
so now we can apply pythagoras theorem
here
finally a b square is equal to b d
square plus a d
square hence a d square will be equal to
a b square minus b d square
so a d square will be equal to a square
minus a by two whole square
a d will be equal to
root 3 a upon 2.
hence oa will be equal to 2 by 380
which is a centroid
centroid property this is the centroid
formula
hence it is given in the question that o
a
which is equal to 20 meter which is the
radius of the circle also
so 20 is equal to 2 by 3
and 80 is root 3 a
upon 2 hence value of a from this will
be equal to
20 root 3 meter
so length of the string of a toy will be
23
meter
if you have any query regarding this you
can drop it in our comment section
and subscribe to lido for more such q a
thank you for watching"

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