Circles | Circles - Exercise 10.4

Question

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.

Solution
Transcript

"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"

Accepted 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.

Circles | Circles - Exercise 10.4

Chapters in Ncert Solutions - Mathematics, Class 9

Exercises in Circles

Question 4