© T Madas O O O O O O O The Circle Theorems. © T Madas 1 st Theorem.

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Presentation transcript:

© T Madas O O O O O O O The Circle Theorems

© T Madas 1 st Theorem

© T Madas The perpendicular bisector of a chord passes through the centre of the circle O

O The perpendicular bisector of a chord passes through the centre of the circle

© T Madas O The perpendicular bisector of a chord passes through the centre of the circle

© T Madas Finding the Centre of Rotation

The shapes below have been produced by rotation. Find the centre of rotation Why does it work?

© T Madas The shapes below have been produced by rotation. Find the centre of rotation

© T Madas The shapes below have been produced by rotation. Find the centre of rotation

© T Madas The shapes below have been produced by rotation. Find the centre of rotation

© T Madas 2 nd Theorem

© T Madas O Inscribed angles which correspond to the same arc are equal Inscribed Angle

© T Madas O Inscribed angles which correspond to the same arc are equal Does this inscribed angle correspond to the same arc?

© T Madas 3 rd Theorem

© T Madas A central angle is twice as large as any inscribed angle which corresponds to the same arc Central Angle Inscribed Angle O

© T Madas Various Forms of the Theorem O O O O O

© T Madas 4 th Theorem

© T Madas O An inscribed angle which corresponds to a diameter ( or semicircle ) is a right angle

5 th Theorem

© T Madas O Cyclic Quadrilateral Opposite angles in a cyclic quadrilateral are supplementary

© T Madas 6 th Theorem

© T Madas O Tangent Tangent point A tangent and a radius drawn at any point on the circumference of the circle meet at right angles

© T Madas 7 th Theorem

© T Madas O The intersection of two tangents to a circle is equidistant from their points of contact. [Their angle of intersection and the central angle formed by the radii at the points of contact, are supplementary]

8 th Theorem

© T Madas O segment sector segment

© T Madas O Alternating Segments

© T Madas O The angle formed by a chord and a tangent at one of its endpoints is equal to the inscribed angle corresponding to the same chord in the alternating segment

Circle Theorem Test

Circle Theorem Mini Test

Practice Question 1

© T Madas O 30° x 45° 30° 15° 150° 15°

© T Madas Practice Question 2

© T Madas 50° z 100° 50° 30° x y O

© T Madas Practice Question 3

© T Madas 70° a b c 20° 70° 20° O

© T Madas Practice Question 4

© T Madas 95° n m 55° 85° 40° p 55° O

© T Madas Practice Question 5

© T Madas 25° x y Tangent point 65° O

© T Madas Practice Question 6

© T Madas 55° s t 110° O

© T Madas Practice Question 7

© T Madas u 28° v 56° O

© T Madas Practice Question 8

© T Madas 300° h O 60° 30° 150°

© T Madas Practice Question 9

© T Madas 130° c 50° 100° O

© T Madas Practice Question 10

© T Madas 50° a b 25° O

© T Madas 50° a b 130° 25° Can you solve this problem without a circle theorem? O

© T Madas Practice Question 11

© T Madas 65° x 230° 115° O

© T Madas Practice Question 12

© T Madas 100° z 200° O

© T Madas Practice Question 13

© T Madas 84° a b O 42° 138°

© T Madas Practice Question 14

© T Madas 32° g O f 148° 32° 64° 296°

© T Madas Practice Question 15

© T Madas 115° p O q 65° 90° 25°

© T Madas Practice Question 16

© T Madas 90° x O 45°

© T Madas Practice Question 17

© T Madas 70° p O A B C AB = BC q r 55° 90° 35° 20°

© T Madas Practice Question 18

© T Madas 72° u O v 90° 18° 72°

© T Madas Practice Question 19

© T Madas 30° a O b c Tangent point 60° 120°

© T Madas Practice Question 20

© T Madas O 58° z y x 32° 58°

© T Madas Practice Question 21

© T Madas 85° x O 95° 85°

© T Madas Practice Question 22

© T Madas 57° t O r 123° 57° Can you think of another reason as to why both these angles are 57° ?

© T Madas Practice Question 23

© T Madas 56° 62° w O x y z 124° 56° 62° 118°

© T Madas Practice Question 24

© T Madas u 45° 160° 155° O 25° 20° 25° 135° v 20°

© T Madas Practice Question 25

© T Madas 30° x O 120° 240°

© T Madas Practice Question 26

© T Madas 75° O x 30° 60°

© T Madas Practice Question 27

© T Madas 72° x O 144° 18°

© T Madas Practice Question 28

© T Madas 40° a O b 140° 50°

© T Madas Practice Question 29

© T Madas 30° θ O 60° 30°

© T Madas Practice Question 30

© T Madas 25° n O 65°

© T Madas Practice Question 31

© T Madas O a 22° b c d Tangent point 22° 68° 56° 124° 68° Exam question

© T Madas