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 each other. Find the length of the string of each phone.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. Find the length of the string of each phone.​

Answer 1

Answer- The above question is from the chapter 'Circles'.

Given question: A circular park of radius 20 m 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. Find the length of the string of each phone. A circular park of radius 20 m 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. Find the length of the string of each phone.​

Solution: Radius of circular park (r) = 20 m

Let the centre be O.

Let the points where Ankur, Syed and David are sitting be represented by the points A, S and D respectively.

Since they are sitting at equal distances, the triangle formed is an equilateral triangle, that can be named as ΔASD.

Since they are sitting at equal distance, AS = AD = SD = 2x.

This distance will be equal to the length of the string of the phone.

Join OD.

OD = r = 20 m

Join OB.

SB = BD = SD/2 = 2x/2 = x m

     |∵ Perpendicular drawn from the centre to the chord bisects it.

In ΔOBD, by Pythagoras Theorem,

OD² = OB² + BD²

20² = OB² + x²

OB² = 400 - x²

$OB = \sqrt{400 - x^2}$

Join AB.

In ΔABD, by Pythagoras Theorem,

AD² = AB² + BD²

(2x)² = AB² + x²

4x² - x² = AB²

AB² = 3x²

AB = √3x

AB = OA + OB

√3x = 20 + $\sqrt{400 - x^2}$

$\sqrt{400 - x^2} = \sqrt3 x - 20$

Squaring both sides, we get,

400 - x² = 3x² + 400 - 40√3 x

40√3 x = 4x²

x = 10√3 m

2x = 20√3 m

∴ The length of the string of each phone = 20√3 m.

Answer 2

Answer:

• The length of the string of each phone - 20√3 m.

Step-by-step explanation:

Given:

• Radius of circular park (r) = 20 m
• Let centre of circle be 'o'
• And tree points are p, q and r.

To Find:

• The length of the string of each phone i.e PQ = QR = RP = ?

Formula used:

• Height of equilateral triangle
• Pythagoras theorem

Now, it is equilateral triangle because its all sides are same.

∴ PQ = QR = RP

Lets each sides of triangle be '2x'.

∴ PQ = QR = RP = 2x

Now, we drawn OM ⊥ QR

⇒ QR = QM + MR

⇒ 2x = QM + MR

∴ QM = MR = x

Now, in ΔPQM

⇒ PQ² = PM² + QM²

⇒ (2x)² = PM² + x²

⇒ 4x² = PM² + x²

⇒ PM² = 4x² - x²

⇒ PM² = 3x²

⇒ PM = √3x

Now, radius (OR) = 20m

Now, In ΔOMR,

⇒ OR² = OM² + MR²

⇒ 20² = OM² + x²

⇒ OM² = 400 - x²

⇒ OM = √400 - x²

Now, PM = PO + OM

⇒ √3x = 20 + √400 - x²

⇒ √3x - 20 = √400 - x²

Now, squaring both the sides,

⇒ (√3x - 20)² = [(√400 - x²)]²

⇒ 3x² - 40√3x + 400 = 400 - x

⇒ 40√3x = 4x²

⇒ x = 10√3

So, 2x = 20√3 m

Hence, the length of the string of each phone - 20√3 m.