Answer:
The triangle formed is Equilateral triangle.
I have taken Perpendicular = P ( you can say M)
√400 - x² = OP (OM)
√3 x = AP (AM)
The length of the string of each person = 20√3 m
Step-by-step explanation:
As given , circular park with radius 20m
Let , Ankur sit on boundary point A
Syed sit on boundary point S
David sit on boundary point D
As given,
Ankur, Syed and David are sitting at equal distance on its boundary
⇒AS = AD = SD
Let AS = AD = SD = 2x
As the sides are equal
⇒ The triangle formed is Equilateral triangle.
Now,
Draw a perpendicular on the side SD
Let OP be the perpendicular
⇒SP = PD = [tex]\frac{1}{2}[/tex]SD = [tex]\frac{1}{2}. 2x = x[/tex]
So, we get
SP = x , PD = x
Now, Draw the line OS and OA
AS , we can see that ,
ΔOPS and ΔAPS are right angle triangle ,
So , apply Pythagoras theorem to ΔOPS and ΔAPS
In ΔOPS
OS² = OP² + PS²
⇒20² = OP² + x²
⇒400 - x² = OP²
⇒√400 - x² = OP
Now,
In ΔAPS
AS² = AP² + PS²
⇒(2x)² = AP² + x²
⇒4x² - x² = AP²
⇒√3x² = AP
⇒√3 x = AP
Now,
We can see that,
AP = AO + OP
⇒√3 x = 20 + √400 - x²
⇒√3 x - 20 + = √400 - x²
Squaring both side
⇒(√3 x - 20 )²= (√400 - x²)²
⇒3x² + 400 - 40√3 x = 400 - x²
⇒3x² + x² = 40√3 x
⇒4x² = 40√3 x
⇒x = 10√3
∴ we get
AS = SD = AD = 2(10√3) = 20√3 m
⇒The length of the string of each person = 20√3 m
