Question

What is the angle subtended by the sector at the centre of a circle? Statement (I): Perimeter of the sector is 16 units.
Statement (II): Arc length of that sector is 10 units.

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

For sector problems, the angle cannot be determined unless enough information is available to calculate both the radius and the arc length. One equation with two unknowns is usually insufficient.

Answer 1

The Correct Option is C

Concept: For a sector of a circle having radius \(r\) and central angle \(\theta\), \[ \text{Perimeter of Sector} = 2r + L \] where \(L\) is the arc length. Also, \[ L=\frac{\theta}{360^\circ}\times 2\pi r \] or equivalently, \[ \theta=\frac{360L}{2\pi r} \] To determine the angle subtended at the centre, both the radius and the arc length must be known.

Step 1: Analyze Statement (I) Alone. Statement (I) states that the perimeter of the sector is 16 units. \[ 2r+L=16 \] This equation contains two unknowns, namely \(r\) and \(L\). For example: \[ r=3,\quad L=10 \] satisfies the condition. Also, \[ r=4,\quad L=8 \] satisfies the same condition. Different values of \(r\) and \(L\) produce different central angles. Therefore, Statement (I) alone is not sufficient.

Step 2: Analyze Statement (II) Alone. Statement (II) states that the arc length is \[ L=10 \] However, the radius is unknown. For instance, \[ r=5 \] and \[ r=10 \] both give different central angles for the same arc length. Thus, Statement (II) alone is also not sufficient.

Step 3: Combine Statements (I) and (II). From Statement (II), \[ L=10 \] Substituting into Statement (I), \[ 2r+10=16 \] \[ 2r=6 \] \[ r=3 \] Now both the radius and arc length are known. Using \[ L=\frac{\theta}{360^\circ}\times 2\pi r \] \[ 10=\frac{\theta}{360^\circ}\times 2\pi(3) \] \[ 10=\frac{\theta}{360^\circ}\times 6\pi \] Hence the value of \(\theta\) can be uniquely determined. Therefore, the two statements together are sufficient. \[ \boxed{\text{Both statements together are sufficient}} \]