Question

The perimeter of sector OAB of a circle with centre O and radius 5.6 cm, is 15.6 cm. Find length of the arc AB. Also find the value of \(\theta\).

Hint:

Express decimal values as fractions to make cancellation straightforward:
\[ 4.4 = \frac{44}{10} \quad \text{and} \quad 5.6 = \frac{56}{10} \] Substituting these into the equation:
\[ \frac{44}{10} = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times \frac{56}{10} \] Allows the fractions and decimals to cancel out perfectly with minimal effort!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Areas Related to Circles, specifically focusing on the perimeter of a sector and its central angle \(\theta\).
A sector of a circle is bounded by two radii and an arc of the circle.
The perimeter of a sector OAB of radius \(r\) is equal to the sum of the lengths of the two bounding radii and the length of the arc:
\[ \text{Perimeter} = OA + OB + \text{Arc Length } AB = 2r + \text{Arc Length } AB \] We are given that the radius \(r = 5.6\) cm and the total perimeter is \(15.6\) cm.
We need to calculate the length of the arc \(AB\) first, and then use the arc length formula to calculate the central angle \(\theta\).

Step 2: Key Formula or Approach:
1. Use the perimeter formula of a sector to find the arc length \(l\):
\[ \text{Perimeter} = 2r + l \implies 15.6 = 2(5.6) + l \] 2. Apply the arc length formula to find the central angle \(\theta\):
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \] We will substitute \(\pi = \frac{22}{7}\) and solve for \(\theta\).

Step 3: Detailed Explanation:

• Find the length of the arc \(AB\) using the perimeter relation:
Let the arc length be \(l\).
\[ \text{Perimeter} = 2r + l \] \[ 15.6 = 2(5.6) + l \] \[ 15.6 = 11.2 + l \] \[ l = 15.6 - 11.2 \] \[ l = 4.4 \text{ cm} \] Thus, the length of the arc \(AB\) is 4.4 cm.

• Set up the formula for the arc length to calculate the central angle \(\theta\):
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \] \[ 4.4 = \frac{\theta}{360^\circ} \times 2 \times \frac{22}{7} \times 5.6 \]

• Simplify the expression on the right-hand side:
Divide 5.6 by 7:
\[ \frac{5.6}{7} = 0.8 \] Substitute this value back into the equation:
\[ 4.4 = \frac{\theta}{360^\circ} \times 2 \times 22 \times 0.8 \] \[ 4.4 = \frac{\theta}{360^\circ} \times 44 \times 0.8 \] \[ 4.4 = \frac{\theta}{360^\circ} \times 35.2 \]

• Solve for \(\theta\):
\[ \theta = \frac{4.4 \times 360^\circ}{35.2} \] Notice that:
\[ \frac{35.2}{4.4} = 8 \] Substitute this back:
\[ \theta = \frac{360^\circ}{8} = 45^\circ \]

Step 4: Final Answer:
The length of the arc \(AB\) is 4.4 cm, and the value of the central angle \(\theta\) is \(45^\circ\).