A piece of thread of length 21.4cm is used to form a sector of a circle of radius 4.2cm on a piece of cloth…

Analyze the question: The thread forms the whole outline of a sector. That means it covers the arc plus the two radii. This is the perimeter of the sector.

Concept: Perimeter of sector = 2r + (θ/360) × 2πr

Solution:

1. Length of the two radii = 2 × 4.2 = 8.4 cm
2. Arc length = 21.4 − 8.4 = 13.0 cm
3. Circumference of the full circle = 2 × (22/7) × 4.2 = 26.4 cm
4. θ/360 = 13.0 / 26.4
5. θ = (13.0 / 26.4) × 360 = 4680 / 26.4 ≈ 177.27°

So θ ≈ 177° to the nearest degree.

Check: Arc = (177/360) × 26.4 ≈ 12.98 cm. Adding 8.4 gives about 21.38 cm, which is approximately 21.4 cm. ✓

Answer: B. 177°.

Tip: When a question says a thread or wire "forms" a sector, check whether it surrounds the whole shape. If so, subtract the 2r for the two straight edges first, then use the arc formula. If you forget the radii and treat 21.4 cm as the arc, you get about 292°, which is not among the options. That is a sign you missed a step.