Proofs Of Theorems 6 Marks out of 150 Theorem Proof PROOF REQUIRED Theorem Proof PROOF REQUIRED Theorem Proof PROOF REQUIRED Theorem Proof PROOF REQUIRED Theorems and Worked Examples เทก1 = 90°; radius bisects chord ๐ Theorem 1 AB=BC ; radius ๊ chord ๐เท 1 = 90°; radius bisects chord AP=PB ; radius ๊ chord WORKED EXAMPLE 1 ( I DO) WORKED EXAMPLE 1 SOLUTION WORKED EXAMPLE 2 (I DO) WORKED EXAMPLE 2 SOLUTION Theorem 7 Theorem 8 PQRO is a cyclic quadrilateral ; opposite angles add up to 180° โ WORKED EXAMPLE 3 (I DO) WORKED EXAMPLE 3 SOLUTION 1 1 2 1 2 2 WORKED EXAMPLE 3 SOLUTION 1 1 2 1 2 2 WORKED EXAMPLE 4 (YOU DO) WORKED EXAMPLE 4 SOLUTION Theorem 9 TANGENT/CHORD THEOREM WORKED EXAMPLE 5 (YOU DO) WORKED EXAMPLE 5 SOLUTION WORKED EXAMPLE 6 (I DO) IF A POINT IS NOT ON THE CIRCLE NOR AT THE CENTRE, THEN YOU MUST USE BASIC TRIANGLE GEOMETRY OR PARALLEL LINES WORKED EXAMPLE 6 SOLUTION UNDERSTANDING SUBTEND เทก MN SUBTENDS ๐ท เทก IS SUBTENDED BY ๐ท MN Theorem 3 ANGLE IN SEMI CIRCLE Given: BD is diameter In only one of the above is the angle subtended by the diameter 90° WORKED EXAMPLE 8 (I DO) WORKED EXAMPLE 8 SOLUTION WORKED EXAMPLE 9 (YOU DO) WORKED EXAMPLE 9 (YOU DO) ARCS AND CHORDS Theorem 2 ANGLE AT CENTRE x 2x ๐ = 22,5° ; ๐๐๐๐๐ ๐๐ก ๐๐๐๐ก๐๐ ๐ = 200° ; ๐๐๐๐๐ ๐๐ก ๐๐๐๐ก๐๐ e ๐ฅ = 42° ; ๐๐๐๐๐ ๐๐ก ๐๐๐๐ก๐๐ ๐ = 70° ; ๐๐๐๐๐ ๐๐ก ๐๐๐๐ก๐๐ WORKED EXAMPLE 10 (YOU DO) ๐บเท = ๐ฅ ; ๐๐๐๐๐ ๐๐ก ๐๐๐๐ก๐๐ เทข1 = ๐ฅ ; alternate angles GJ // KH ๐ป ๐ฝเทก1 + ๐ฝเทก2 = ๐ฅ ; tan/chord theorem เทข1 + ๐ป เทข2 = 90° ; tan ๊ ๐๐๐๐๐ข๐ ๐ป ∴เทฃ ๐ป2 = 90° − ๐ฅ เทข3 + ๐ฝเทก1 = 180°-2x ; angles of a โ ๐ป But OJ = OH ; both radii เทข3 = 90° − ๐ฅ ∴เทข๐ฝ1 + ๐ป WORKED EXAMPLE 11 (I DO) WORKED EXAMPLE 11 (SOLUTION) WORKED EXAMPLE 12 (YOU DO) WORKED EXAMPLE 12 SOLUTION PROVING DIAMETERS WORKED EXAMPLE 13 (I DO) WORKED EXAMPLE 13 SOLUTION WORKED EXAMPLE 14 (YOU DO) WORKED EXAMPLE 14 SOLUTION (YOU DO) ANGLES IN SAME SEGMENT Theorem 4 WORKED EXAMPLE 15 (I DO) WORKED EXAMPLE 16 (YOU DO) WORKED EXAMPLE 16 SOLUTION WORKED EXAMPLE 17 (I DO) WORKED EXAMPLE 18 (YOU DO) WORKED EXAMPLE 19 (I DO) WORKED EXAMPLE 20 (YOU DO) PRACTICE EXAMPLE 1 เท ๐ด๐๐ถ PRACTICE EXAMPLE 2 PRACTICE EXAMPLE 3 PRACTICE EXAMPLE 4 PRACTICE EXAMPLE 5 PRACTICE EXAMPLE 6 PRACTICE EXAMPLE 7 PRACTICE EXAMPLE 8 PRACTICE EXAMPLE 9 PRACTICE EXAMPLE 10