Circle Theorems Euclid of Alexandria Circa 325 265

Circle Theorems Euclid of Alexandria Circa 325 - 265 BC O The library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained 600 000 manuscripts.

A Reminder about parts of the Circle Ta ng Major Arc en Circumference t radius Major Segment diameter t n e ng Ta chord Minor Segment Major Sector Minor Arc Minor Sector Parts Tangent

Introductory Terminology yo yo o o xo xo A Term’gy B B B A Arc AB subtends angle x at the centre. Arc AB subtends angle y at the circumference. Chord AB also subtends angle x at the centre. Chord AB also subtends angle y at the circumference. A o xo yo

Th 1 Theorem 1 Measure the angles at the centre and circumference and make a conjecture. xo xo o o yo yo xo yo o xo xo xo o yo

Theorem 1 The angle subtended at the centre of a circle (by an arc or chord) is twice the angle subtended at the circumference by the same arc or chord. (angle at centre) Measure the angles at the centre and circumference and make a conjecture. xo xo o o 2 xo xo o 2 xo Angle x is subtended in the minor segment. o 2 xo xo xo o 2 xo Watch for this one later. o 2 xo

Example Questions Find the unknown angles giving reasons for your answers. 1 2 xo o 35 o yo 84 o A B angle x = angle y = o A 42 o (Angle at the centre). 70 o(Angle at the centre) B

Example Questions Find the unknown angles giving reasons for your answers. 3 4 yo 62 o o o po xo A B qo 42 o B A angle x = (180 – 2 x 42) = 96 o (Isos triangle/angle sum triangle). angle y = 48 o (Angle at the centre) angle p = 124 o (Angle at the centre) angle q = (180 – 124)/2 = 280 (Isos triangle/angle sum triangle).

Theorem 2 The angle in a semi-circle is a right angle. This is just a special case of Theorem 1 and is referred to as a theorem for convenience. o Diameter Find the unknown angles below stating a reason. a 30 o c 90 o angle in a semi-circle angle b = 90 o angle in a semi-circle angle c = angle sum triangle 20 o angle d = 90 o angle in a semi-circle angle e = 60 o angle sum triangle d o angle a = e 70 o b Th 2

Theorem 3 xo Angles subtended by an arc or chord in the same segment are equal. yo xo xo yo xo Th 3 xo

Theorem 3 Angles subtended by an arc or chord in the same segment are equal. Find the unknown angles in each case 38 o yo xo 30 o 40 o yo Angle x = angle y = 38 o xo Angle x = 30 o Angle y = 40 o

The angle between a tangent and a radius is 90 o. (Tan/rad) Theorem 4 o Th 4

The angle between a tangent and a radius is 90 o. (Tan/rad) o Theorem 4

If OT is a radius and AB is a tangent, find the unknown angles, giving reasons for your answers. 30 o o xo yo 36 o B zo T A angle x = 180 – (90 + 36) = 54 o Tan/rad angle sum of triangle. angle y = 90 o angle in a semi-circle angle z = 60 o angle sum triangle

Theorem 5 The Alternate Segment Theorem. The angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment. Find the missing angles below giving reasons in each case. xo yo yo xo o 60 xo yo zo o 45 angle x = 45 o (Alt Seg) angle y = 60 o (Alt Seg) angle z = 75 o angle sum triangle Th 5

Theorem 6 Cyclic Quadrilateral Theorem. The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180 o) x Th 6 y w p s z r q Angles x + w = 180 o Angles p + q = 180 o Angles y + z = 180 o Angles r + s = 180 o

Theorem 6 Cyclic Quadrilateral Theorem. The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180 o) x y 110 o Find the missing angles below given reasons in each case. angle y = q p 85 o angle x = 180 – 85 = 95 o r 70 o (cyclic quad) 180 – 110 = 70 o (cyclic quad) 135 o angle p = 180 – 135 = 45 o (straight line) angle q = 180 – 70 = 110 o (cyclic quad) angle r = 180 – 45 = 135 o (cyclic quad)

Theorem 7 Two Tangent Theorem. From any point outside a circle only two tangents can be drawn and they are equal in length. R P Q Q U T T P R PT = PQ Th 7 PT = PQ U

Theorem 7 Two Tangent Theorem. From any point outside a circle only two tangents can be drawn and they are equal in length. PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. Q yo xo O 98 o angle w = 90 o (tan/rad) angle x = 90 o (tan/rad) angle y = zo P wo T angle z = 49 o (angle at centre) 360 o – 278 = 82 o (quadrilateral)

Theorem 7 Two Tangent Theorem. From any point outside a circle only two tangents can be drawn and they are equal in length. PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. zo Q yo O angle w = 90 o (tan/rad) xo 80 o P wo T angle x = 180 – 140 = 40 o 50 o (angles sum tri) angle y = 50 o (isos triangle) angle z = 50 o (alt seg)

Theorem 8 Chord Bisector Theorem. A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord. . Find length OS O O 3 cm S Th 8 8 cm T OS = 5 cm (pythag triple: 3, 4, 5)

Theorem 8 Chord Bisector Theorem. A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord. . Find angle x O O 22 o S xo T U Angle SOT = 22 o (symmetry/congruenncy) Angle x = 180 – 112 = 68 o (angle sum triangle)

Mixed Questions U PTR is a tangent line to the circle at T. Find angles SUT, SOT, OTS and OST. O S R 65 o T Angle SUT = 65 o (Alt seg) P Angle SOT = 130 o (angle at centre) Mixed Q 1 Angle OTS = 25 o (tan rad) Angle OST = 25 o (isos triangle)

Mixed Questions Q PR and PQ are tangents to the circle. Find the missing angles giving reasons. U y 110 o P O z Mixed Q 2 w x 48 o R Angle w = 22 o (cyclic quad) Angle x = 68 o (tan rad) Angle y = 44 o (isos triangle) Angle z = 68 o (alt seg)

The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended at the circumference by the same arc or chord. A O B C Worksheet 3 Theorem 1 and 2

Angles subtended by an arc or chord in the same segment are equal. A D O B C Theorem 3 Worksheet 4

The angle between a tangent and a radius drawn to the point of contact is a right angle. O A T Theorem 4 B Worksheet 5

The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment. D B C O T A Theorem 5 Worksheet 6

The opposite angles of a cyclic quadrilateral are supplementary (Sum to 180 o). B A C D Theorem 6

The two tangents drawn from a point outside a circle are of equal length. Theorem 7 A O P B Theorem 8 O A B C A line, drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord. Worksheet 8