Read the parts of a circle correctly
A circle is the locus of points at a fixed distance from a fixed centre. That fixed distance is the radius; a diameter passes through the centre and is twice the radius. A chord joins any two points on the circle. An arc is part of the circumference, a sector is enclosed by two radii and an arc, and a segment is enclosed by a chord and an arc. A tangent touches the circle at a point, while a secant cuts it at two points.
Key definition
Arc
Part of the circumference, a sector is enclosed by two radii and an arc, and a segment is enclosed by a chord and an arc.
Exam keywords:circumferencesectorenclosedsegment
A diagram's appearance is not proof that a line is a diameter or a tangent. Use the information stated or marked. If O is the centre and A and B lie on the circle, so triangle OAB is isosceles. This elementary observation supports many circle-angle solutions. Label the relevant centre, chord and arc before selecting a theorem, and distinguish angles standing on the same chord from angles standing on opposite sides of that chord.
Angles at the centre and circumference
The angle subtended by an arc at the centre is twice the angle subtended by the same arc at a point on the remaining circumference. If central angle AOB is 100° and point C is on the major arc AB, angle ACB is 50°. Specify the same arc: choosing a point on the other arc changes which central reflex angle is relevant. In particular, an angle subtended by a diameter at the circumference is 90°, because the matching central angle is 180°.
Angles in the same segment are equal: if two angles at the circumference stand on the same chord and their vertices lie on the same side of it, their angles are equal. This gives a direct link between separated parts of a diagram. If A, B, C and D lie on a circle and C and D are on the same side of chord AB, then angle. Do not apply the result to arbitrary angles that merely happen to use points from the same circle.
Cyclic quadrilaterals and tangents
A cyclic quadrilateral has all four vertices on a circle. Opposite interior angles total 180°. Thus if one angle is 112°, its opposite is 68°. An exterior angle of a cyclic quadrilateral equals the opposite interior angle. These properties can also help show that a quadrilateral is cyclic when the relevant conditions hold. Adjacent angles are not generally supplementary unless an additional condition, such as parallel sides, makes them so.
A radius drawn to the point of contact is perpendicular to a tangent. Two tangent segments from the same external point have equal lengths. If PA and PB are tangents, then ⟂ PA and OB ⟂ PB. The alternate segment theorem says the angle between a tangent and a chord equals the angle in the opposite segment subtended by that chord. Identify the exact tangent-chord angle and the corresponding inscribed angle, rather than selecting whichever angle looks closest in a sketch.
Chords and intersecting lines
A perpendicular from the centre to a chord bisects the chord. Conversely, the line from the centre to a chord's midpoint is perpendicular to the chord. If radius is 13 cm and the centre is 5 cm from a chord, half the chord is √(13² − 5²) = 12 cm, so the full chord is 24 cm. Equal chords are equally distant from the centre, and equal chords subtend equal central angles within the same circle or congruent circles.
When two chords intersect inside a circle at P, the products of their segment lengths are equal, where A and B are the endpoints of one chord and C and D of the other. If PA = 3, PB = 8 and PC = 4, then PD = 6. Distinguish this internal-chord rule from the external secant rule, which uses an external segment multiplied by the whole secant length. Always label the segments before substitution.
Construct with an unmarked ruler and compasses
A construction records a geometric method, not just a measured-looking diagram. To construct the perpendicular bisector of AB, draw equal-radius arcs centred at A and B with radius greater than half AB. Let their intersections be P and Q. Joining P and Q gives the required line because both points are equidistant from A and B. Leave the construction arcs visible. Measuring the midpoint with a ruler and drawing a line that looks perpendicular does not demonstrate the same construction.
To bisect an angle, draw an arc from its vertex crossing the arms at P and Q. With equal radii centred at P and Q, draw intersecting arcs inside the angle. Join their intersection to the vertex. To construct 60°, construct an equilateral triangle; bisect 60° to obtain 30°. Construct a perpendicular to obtain 90°, then bisect it to obtain 45°. Combining a 30° and a 45° angle can give 75°. A paper construction still needs suitable tools and personal practice; reading the steps alone does not prove drawing accuracy.
Interpret a locus as all valid points
The locus of points a fixed distance r from point O is a circle. The locus equidistant from two distinct points A and B is their perpendicular bisector. Points equidistant from two intersecting lines lie on their two angle bisectors. Points at a fixed perpendicular distance from a straight line lie on two parallel lines, one on each side. If two conditions must hold simultaneously, construct both loci and take their intersections; those intersection points satisfy both requirements.
For example, a point must be 4 cm from A and equidistant from A and B. Draw the circle centre A radius 4 cm and the perpendicular bisector of AB. Their intersections, if any, are the permitted points. Depending on AB, there may be two, one or no intersections. A locus can also describe a region if the wording says “less than” a distance. Use a strict or included boundary consistently. Self-check: a cyclic quadrilateral has opposite angles x and 73°, so x = 107°. A diameter subtends a right angle at the circumference. State the theorem after each result and keep construction arcs available for inspection.
Separate external length products
If an external point P has a tangent PT and a secant meeting the circle first at A then at B, then. Here PB is the whole secant length from P, not just the inside chord AB. For PA = 3 cm and PB = 12 cm, PT = 6 cm. The inside chord is 9 cm. Two secants from the same external point satisfy the analogous equality of external times whole lengths. Mark the outside point and endpoints before choosing a product rule.