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Home , Chord (geometry)

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 푁෡ = 90° 1 ; radius bisects chord Theorem 1 AB=BC ; radius ꓕ chord

푃෠1 = 90°; radius bisects chord AP=PB ; radius ꓕ 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 ; opposite add up to 180°

● WORKED EXAMPLE 3 (I DO) WORKED EXAMPLE 3 SOLUTION

2 1 1 2

1 2 WORKED EXAMPLE 3 SOLUTION

2 1 1 2

1 2 WORKED EXAMPLE 4 (YOU DO) WORKED EXAMPLE 4 SOLUTION Theorem 9 /CHORD THEOREM WORKED EXAMPLE 5 (YOU DO) WORKED EXAMPLE 5 SOLUTION WORKED EXAMPLE 6 (I DO)

IF A POINT IS NOT ON THE NOR AT THE CENTRE, THEN YOU MUST USE BASIC TRIANGLE GEOMETRY OR PARALLEL LINES WORKED EXAMPLE 6 SOLUTION UNDERSTANDING SUBTEND

MN SUBTENDS 푷෡ 푷෡ IS SUBTENDED BY MN Theorem 3 IN SEMI CIRCLE

Given: BD is

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

푐 = 22,5° ; 푎푛푔푙푒 푎푡 푐푒푛푡푟푒 푑 = 200° ; 푎푛푔푙푒 푎푡 푐푒푛푡푟푒

e

푥 = 42° ; 푎푛푔푙푒 푎푡 푐푒푛푡푟푒 푒 = 70° ; 푎푛푔푙푒 푎푡 푐푒푛푡푟푒 WORKED EXAMPLE 10 (YOU DO)

퐺෠ = 푥 ; 푎푛푔푙푒 푎푡 푐푒푛푡푟푒 퐻෢1 = 푥 ; alternate angles GJ // KH 퐽෡1 + 퐽෡2 = 푥 ; tan/chord theorem

퐻෢1 + 퐻෢2 = 90° ; tan ꓕ 푟푎푑푖푢푠 ∴෣퐻2 = 90° − 푥

퐻෢3 + 퐽෡1 = 180°-2x ; angles of a ∆ But OJ = OH ; both radii ∴෢퐽1 + 퐻෢3 = 90° − 푥 WORKED EXAMPLE 11 (I DO) WORKED EXAMPLE 11 (SOLUTION) WORKED EXAMPLE 12 (YOU DO) WORKED EXAMPLE 12 SOLUTION PROVING 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 퐴푂෠퐶 PRACTICE EXAMPLE 2

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PRACTICE EXAMPLE 10