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Circles are all around us. They are products of both our natural environment and of our artificial environment.

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Circles are part of the astronomically big Whirlpool Galaxy Model of an Atom Throughout history circles have been considered to have a certain perfection and harmony to them. Even today there is a certain mystic and wonder around the circle. and the infinitesimally small.

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CIRCLE TERMINOLOGY To study circles, it is important that we are familiar with the terminology of circles.

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O A circle is defined by a group of points all being the same distance from a point identified as the centre, O. A circle defines three distinct groups of points: - the points inside the circle (including centre). - the points on the circle. - the points outside the circle.

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O Radius – a line segment drawn from the centre to any point on the circle. Radius A C B Chord – a line segment between any two points on a circle Note: A line segment is a line which has a beginning and an end (the endpoints) and consequently it has a defined length and can be measured. When naming line segments, the two endpoints are always used in either order. OA BC

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O Diameter – a chord that passes through the centre of a circle. D E DE

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Arc – a series of points on the circumference between two given points on the circumference. Arcs are classified into two groups: Minor arcs – that make up less than one half of the circumference. Major arcs – that make up one half or more of the circumference. 3 letters must be used to name a major arc (two letters to identify the ends of the arc and a third letter between them to identify any other point on the arc). E D F B A M DFEEFD ornot DEAMB or AB When naming arcs, Minors arcs can be named using 2 letters representing the endpoints (although the 3 letter designation can also be used as with major arcs).

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Line – a series of points along a straight line that go infinitely in both directions. To name a line you can use any two points on the line L M N Secant – a straight line that intersects a circle at two points. AB LNNLorMNor Tangent – a straight line that intersects a circle at one point. BAorBTorATor Point of Tangency – point of intersection which is shared by the tangent line and the circle. T A B TT

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A B DISTINCTION BETWEEN CHORDS AND ARCS A B A B Chords Just as Criminals Steal Apples so do However it does NOT make sense to say that: In the same way it doesn’t make sense to say that: Arcs Subtend Chords Apples Steal Criminals AB subtendstherefore AB butdoes not subtend SubtendArcs A B

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When referring to chords, radii, arcs, diameters, secants and tangents, it is important to use the correct symbol above the letter names. - used for line segments (chords, radii or diameters) - used for lines (tangents or secants) - used for arcs (minor or major) A AB The 3 geometric items above are all referring to different sets of points. B AB - refers to all points on the circumference between A and B - refers to all points on a straight line between A and B - refers to all points on a straight line between and beyond A and B

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Angles are also elements of a circle that are often referred to. An angle is a rotational separation between two rays. The angles of a circle are classified into 4 categories based on the location of their vertices: (1) Central angle – vertex at the centre (2) Interior angle – vertex inside the circle (3) Exterior angle – vertex outside the circle (4) Inscribed angle – vertex on the circumference

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B O A AOB is a central angle because its vertex, O, is at the centre of the circle. Incidentally, it is also an interior angle because its vertex is inside the circle. However, just as we don’t refer to a square as a rectangle (eventhough it is) so do we not refer to a central angle as an interior angle. AOB intersects AB -formed by two radii

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O DEF is an interior angle because its vertex, E, is inside of the circle. F D E G H DEF intersects and its vertical angle, GEH intersects DF GH -formed by two chords

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O M I K L J N IKM, IKN and MKN are all exterior angles because their vertex, K, is outside of the circle. IKM intersects andIMJL -formed by two secants, two tangents or a secant and a tangent MN MKN intersects and LN IN IKN intersects and JN

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O QPT, QPR and RPT are all inscribed angles because their vertex, P, is on the circumference of the circle. R Q P T S QPT intersects QPR intersects RPT intersects QRP QR RSP -formed by two secants or a secant and a tangent

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КОНЕЦ final finito The end le fin sof ﭙﺎﻴﺎﻥ

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Circles Chapter 12.

Circles Chapter 12.

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