Question

PQRS is a diameter of a circle whose radius is r. The lengths of PQ, QR and RS are equal Semi-circles are drawn on PQ and QS to create the shaded figure below : The perimeter of the shaded figure is​

Answer 1

Answer:

r(7π + 6)/3

Step-by-step explanation:

We will use the concept of semi perimeter to solve the required question.

Semi perimeter = πr where r is radius

Given that radius of circle is r.

⇒ PS = diameter = 2r

⇒ PQ + QR + RS = 2r

⇒ PQ + PQ + PQ = 2r

⇒ 3PQ = 2r

⇒ PQ = 2r/3

∴ PQ = QR = RS = 2r/3

The required perimeter of shaded region includes the following parts:

• Length of PS
• Semi perimeter of circle whose diameter is QS.
• Semi perimeter of circle whose diameter is PQ.

Semi perimeter of circle through QRS:

⇒ π × (Radius)

⇒ π × QR

⇒ π × 2r/3

⇒ 2πr/3 __________(1)

Semi perimeter of circle through PQ:

⇒ π × (Radius)

⇒ π × (PQ/2)

⇒ π × (2r/3 ÷ 2)

⇒ π × r/3

⇒ πr/3 ____________(2)

Required perimeter is given by:

➝ Equation (1) + Equation (2) + PS

➝ 2πr/3 + πr/3 + 2r

➝ (6πr + πr + 6r)/3

➝ (7πr + 6r)/3

➝ r(7π + 6)/3

This is the required answer.