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 each other. name the type of triangle asd formed Find the length of AM

Find the length of OM

Find the length of the string of each person

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 = SD =

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

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

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