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
෡1 = 90°; radius bisects chord
𝑁
Theorem 1
AB=BC ; radius ꓕ chord
𝑃෠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 cyclic quadrilateral ; opposite angles add up to 180°
●
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
෡
MN SUBTENDS 𝑷
෡ IS SUBTENDED BY
𝑷
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
𝑐 = 22,5° ; 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑒
𝑑 = 200° ; 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑒
e
𝑥 = 42° ; 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑒
𝑒 = 70° ; 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑒
WORKED EXAMPLE 10
(YOU DO)
𝐺෠ = 𝑥
; 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑒
෢1 = 𝑥 ; alternate angles GJ // KH
𝐻
𝐽෡1 + 𝐽෡2 = 𝑥 ; tan/chord theorem
෢1 + 𝐻
෢2 = 90° ; tan ꓕ 𝑟𝑎𝑑𝑖𝑢𝑠
𝐻
∴෣
𝐻2 = 90° − 𝑥
෢3 + 𝐽෡1 = 180°-2x ; angles of a ∆
𝐻
But OJ = OH ; both radii
෢3 = 90° − 𝑥
∴෢𝐽1 + 𝐻
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
෠
𝐴𝑂𝐶
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