Circles and Geometric Constructions
Circles and Geometric Constructions
Circles supply 1 to 2 questions in the Part-III paper, usually on chord and tangent properties or on the angle subtended at the centre. Constructions are asked as a description of steps rather than a drawing, so learn the sequence and the tool used at each stage.
Parts of a Circle
| Part | Meaning |
|---|---|
| Radius | Centre to any point on the circle |
| Diameter | Chord through the centre; the longest chord |
| Chord | Segment joining two points on the circle |
| Arc | Part of the circumference |
| Sector | Region between two radii and an arc |
| Segment | Region between a chord and an arc |
| Tangent | A line touching the circle at exactly one point |
| Secant | A line cutting the circle at two points |
The sector and segment pair is regularly reversed. A sector is bounded by two radii, like a slice of cake; a segment is cut off by a chord.
✗ A segment is the region between two radii and an arc | ✓ That is a sector; a segment is bounded by a chord and an arc
Chord and Tangent Properties
- A perpendicular from the centre to a chord bisects the chord, and the converse holds.
- Equal chords are equidistant from the centre, and chords equidistant from the centre are equal.
- The longer the chord, the nearer it lies to the centre; the diameter is the longest chord and passes through it.
- A tangent is perpendicular to the radius at the point of contact.
- The two tangents drawn from an external point are equal in length.
- Only one tangent can be drawn at a given point on a circle.
Angles in a Circle
Angle at the centre = 2 × angle at the circumference on the same arc
- The angle in a semicircle is always 90 degrees, which is a direct consequence of the rule above.
- Angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral are supplementary, adding to 180 degrees.
- The exterior angle of a cyclic quadrilateral equals the interior opposite angle.
The semicircle result is the one most often asked, usually dressed up as a triangle drawn on a diameter.
Constructions
Constructions use only a ruler and compass. A protractor is not permitted in a proper construction, and a question sometimes asks which angles can be constructed without one.
| Construction | Method in short |
|---|---|
| Perpendicular bisector | Arcs of equal radius from both ends, join intersections |
| Angle bisector | Arc from the vertex, then equal arcs from the two cut points |
| 60° angle | One arc from the vertex, same radius cut on the arc |
| 30°, 15° | Bisect 60°, then bisect again |
| 90°, 45° | Construct 60° and 120°, bisect between; bisect 90° for 45° |
The angles constructible with compass alone are the multiples and repeated bisections of 60 and 90 degrees — so 15, 30, 45, 75 and 105 are all possible, while 20 and 50 are not. That last point is a genuine question.
अर्द्धवृत्त में बना कोण हमेशा 90° होता है।
TRE pointer: The angle in a semicircle and the cyclic quadrilateral rule together account for most circle questions in a teaching paper. Constructions are asked in words, so know which tool does what — and remember that 60 and 90 degrees are the only angles built directly, everything else coming from bisection. With five options and −1/3 negative marking, sketch the circle before answering; a chord drawn wrongly in the mind costs the mark.
60-Second Recap
- Sector is bounded by two radii; segment is bounded by a chord.
- A perpendicular from the centre bisects a chord; equal chords are equidistant from the centre.
- A tangent is perpendicular to the radius at the point of contact.
- Tangents from an external point are equal in length.
- Angle at the centre is twice the angle at the circumference; the angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral are supplementary.
Frequently Asked Questions
What is the angle in a semicircle?
Always ninety degrees. It follows from the rule that the angle at the centre is twice the angle at the circumference on the same arc: the diameter subtends a straight angle of a hundred and eighty at the centre, so the angle at the circumference is ninety.
What is the difference between a sector and a segment?
A sector is bounded by two radii and the arc between them, like a slice of cake. A segment is the region cut off by a chord and the arc above it. The two names are reversed by candidates so often that both appear as options in almost every such question.
What is the relationship between a tangent and a radius?
The tangent is perpendicular to the radius drawn to the point of contact. Also, two tangents drawn to a circle from the same external point are equal in length, and only one tangent can be drawn at any given point on the circle.
Which angles can be constructed with a compass alone?
Sixty and ninety degrees are constructed directly, and everything else comes from bisecting or combining them: thirty, forty-five, fifteen, seventy-five and a hundred and five are all possible. Angles such as twenty or fifty degrees cannot be constructed without a protractor.