CIRCLES
Frequently Asked Questions
A circle is a set of all points in a plane that are at a fixed distance (radius) from a fixed point called the centre.
The distance from the centre to any point on the circle. All radii of a circle are equal.
A line segment passing through the centre and touching both ends of the circle. It is twice the radius.
\( \text{Diameter} = 2 \times \text{Radius} \).
A chord is any line segment joining two points on a circle.
The diameter is the longest chord.
A part of the circumference between two points is called an arc.
The smaller arc between two points on a circle.
The larger arc between the same two points on a circle.
A \(180^\circ\) arc formed by endpoints of the diameter.
A region bounded by a chord and its corresponding arc.
Minor segment and major segment.
A region enclosed by two radii and the connecting arc.
The angle subtended at the centre by an arc or chord.
Angle formed at centre by joining centre with chord endpoints.
Equal chords subtend equal angles at the centre.
If two chords subtend equal angles at the centre, then the chords are equal.
Perpendicular from centre to chord bisects the chord.
If a line from centre bisects a chord, it is perpendicular to the chord.
Equal chords are equidistant from the centre.
Chords equidistant from centre are equal in length.
A quadrilateral whose all vertices lie on a single circle.
Always supplementary:
Exterior angle = interior opposite angle.
Square, rectangle, isosceles trapezium.
The perpendicular bisector of any chord passes through the centre.
Check if opposite angles sum to \(180^\circ\).
Angle formed on the circle's circumference by two chords.
Angles in the same segment are equal.
Angle in a semicircle is always \(90^\circ\).
Wheels, rings, clocks, gears, coins, beads, traffic roundabouts, engineering drawings.
Construction, design, architecture, astronomy, trigonometry, physics, map creation.
Rainbows, bridges, arches, domes, curved roads.
Only three non-collinear points uniquely determine a circle.
Infinitely many.
Circles having the same centre but different radii.
A line touching a circle at exactly one point.
Two tangents can be drawn (next chapter concept).
Fix compass at centre, adjust radius, rotate around point.
Use compass arcs from both chord endpoints.
Draw perpendicular bisectors of any two chords; they meet at centre.
Yes, in 0, 1, or 2 points.
Chord shared by two circles when they intersect.
When distance between centres < difference of radii.
Distance between centres = sum of radii.
Distance between centres = difference of radii.
Use congruent triangles formed by joining chord endpoints to centre.
Based on the theorem: angle in semicircle = \(90^\circ\).
If opposite angles add up to \(180^\circ\).
No, circle has no straight sides.
No, circle has no vertices.
Yes, always.
Midpoint of chord lies on perpendicular bisector from centre.
Opposite angles sum to \(180^\circ\).
Only chord that passes through centre.
It lies exactly at the geometric centre.
\(360^\circ\).
Chord is straight; arc is curved.
Use: arc measure = central angle (in degrees).
Several, but their sum is always \(360^\circ\).
Inscribed angle is half the measure of central angle.
Yes, directly linked.
Because they subtend arcs that together cover \(360^\circ\).
Yes, but only if it is a rectangle.
Yes, every rectangle can be inscribed in a circle.
Yes, all squares are cyclic quadrilaterals.
No—only rhombuses with equal opposite angles.
Check angle properties or perpendicular bisector intersection.
When angles in the same segment are equal.
Right angle \((90^\circ)\).
