Properties and constructions: including: tangent, arc, sector and segment

The idea

This row requires you to know and use the four named parts of a circle: tangent, arc, sector and segment. A tangent is a straight line that touches the circle at exactly one point, called the point of contact, and is perpendicular to the radius drawn to that point. An arc is a connected part of the circumference. A sector is the region bounded by two radii and the arc between them, like a slice of a cake. A segment is the region bounded by a chord and the arc cut off by that chord. Distinguish carefully: a sector uses two radii, while a segment uses a chord. A diameter is a special chord through the centre, and a semicircle is a special sector and segment.

What earns the marks

• A tangent touches the circle at exactly one point and is perpendicular to the radius at that point of contact.
• An arc is a part of the circumference of the circle.
• A sector is the region enclosed by two radii and the arc between them.
• A segment is the region enclosed by a chord and the arc cut off by that chord.
• A diameter is a chord passing through the centre; a semicircle is a special sector and segment.

Common mistakes and how to put them right

• Confusing a sector with a segment.

  Put it right: A sector is bounded by two radii and an arc, whereas a segment is bounded by a chord and an arc.

• Thinking a tangent crosses the circle at two points.

  Put it right: A tangent touches at exactly one point, the point of contact.

• Calling any line across the circle a diameter.

  Put it right: A diameter must pass through the centre; other chords do not.

• Believing an arc must be shorter than a semicircle.

  Put it right: An arc can be minor or major, so it may be more than half the circumference.

Exam tips

• Read the diagram labels carefully and check whether the region is bounded by radii or by a chord before choosing sector or segment.
• Remember the tangent–radius right angle at the point of contact; it is often the key to identifying a tangent in a diagram.
• For multiple-choice questions, eliminate options by testing the boundary: two radii means sector, one chord means segment.