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Circles i

  1. Find the value of x :

Theorem 4 The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circumference of the circle.

Proof :

Consider a circle, with centre O and with A and B on the circumference. Draw a chord A B . Draw radii O A and O B . Select any point P on the circumference of the circle. Draw lines P A and P B . Draw P O and extend to R . The aim is to prove that A O B ^ = 2 · A P B ^ . A O R ^ = P A O ^ + A P O ^ (exterior angle = sum of interior opp. angles) But, P A O ^ = A P O ^ ( A O P is an isosceles ) A O R ^ = 2 A P O ^ Similarly, B O R ^ = 2 B P O ^ . So,

A O B ^ = A O R ^ + B O R ^ = 2 A P O ^ + 2 B P O ^ = 2 ( A P O ^ + B P O ^ ) = 2 ( A P B ^ )

Circles ii

  1. Find the angles ( a to f ) indicated in each diagram:

Theorem 5 The angles subtended by a chord at the circumference of a circle on the same side of the chord are equal.

Proof :

Consider a circle, with centre O . Draw a chord A B . Select any points P and Q on the circumference of the circle, such that both P and Q are on the same side of the chord. Draw lines P A , P B , Q A and Q B . The aim is to prove that A Q B ^ = A P B ^ .

A O B ^ = 2 A Q B ^ at centre = twice at circumference and AOB ^ = 2 A P B ^ at centre = twice at circumference 2 A Q B ^ = 2 A P B ^ A Q B ^ = A P B ^

Theorem 6 (Converse of Theorem [link] ) If a line segment subtends equal angles at two other points on the same side of the line, then these four points lie on a circle.

Proof :

Consider a line segment A B , that subtends equal angles at points P and Q on the same side of A B . The aim is to prove that points A , B , P and Q lie on the circumference of a circle. By contradiction. Assume that point P does not lie on a circle drawn through points A , B and Q . Let the circle cut A P (or A P extended) at point R .

A Q B ^ = A R B ^ 's on same side of chord but AQB ^ = A P B ^ ( given ) A R B ^ = A P B ^ but this cannot be true since ARB ^ = A P B ^ + R B P ^ ( ext. of )

the assumption that the circle does not pass through P , must be false, and A , B , P and Q lie on the circumference of a circle.

Circles iii

  1. Find the values of the unknown letters.

Cyclic quadrilaterals

Cyclic quadrilaterals are quadrilaterals with all four vertices lying on the circumference of a circle. The vertices of a cyclic quadrilateral are said to be concyclic .

Theorem 7 The opposite angles of a cyclic quadrilateral are supplementary.

Proof :

Consider a circle, with centre O . Draw a cyclic quadrilateral A B P Q . Draw A O and P O . The aim is to prove that A B P ^ + A Q P ^ = 180 and Q A B ^ + Q P B ^ = 180 .

O 1 ^ = 2 A B P ^ 's at centre O 2 ^ = 2 A Q P ^ 's at centre But, O 1 ^ + O 2 ^ = 360 2 A B P ^ + 2 A Q P ^ = 360 A B P ^ + A Q P ^ = 180 Similarly , QAB ^ + QPB ^ = 180

Theorem 8 (Converse of Theorem [link] ) If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

Proof :

Consider a quadrilateral A B P Q , such that A B P ^ + A Q P ^ = 180 and Q A B ^ + Q P B ^ = 180 . The aim is to prove that points A , B , P and Q lie on the circumference of a circle. By contradiction. Assume that point P does not lie on a circle drawn through points A , B and Q . Let the circle cut A P (or A P extended) at point R . Draw B R .

Q A B ^ + Q R B ^ = 180 opp. 's of cyclic quad but QAB ^ + QPB ^ = 180 ( given ) Q R B ^ = Q P B ^ but this cannot be true since QRB ^ = Q P B ^ + R B P ^ ( ext . of )

the assumption that the circle does not pass through P , must be false, and A , B , P and Q lie on the circumference of a circle and A B P Q is a cyclic quadrilateral.

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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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