急!急! F4 Properties Circles q4

5.
Denote X as the intersecting point of AC and BD.

∠CAD = ∠DBC (∠s in the same segment)
∠CAD = 18°

In ΔACP :
∠ACB = ∠CAD + ∠DPC (ext. ∠ of Δ)
∠ACB = 18° + 42°
∠ACB = 60°

In ΔXBC :
∠AXB = ∠ACB + ∠DBC (ext. ∠ of Δ)
∠AXB = 60° + 18°
∠AXB = 78°


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7.
(a)
BC is a chord.
Then ∠BOC < 180°

∠ at circumference =∠ at centre / 2
x = ∠BOC/2
x < 180°/2
Hence, 0 < x < 90°

(b)
Equal arcs subtend equal chords.
Since arc AB = arc AC, then AB = AC

In ΔABC :
∠ACB = ∠ABC (equal sides to equal ∠s)
∠ACB + ∠ABC + ∠BAC = 180° (∠ sum of Δ)
∠ACB + ∠ACB + 38° = 180°
∠ACB = 71°


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9.
(a)
∠AOC = 2∠OBC (∠at centre = 2∠ at circum.)
∠AOC = 2(22°)
∠AOC = 44°

∠AOC : ∠COD = arc AC : arc CD
44° : ∠COD = 2 : 3
∠COD = 66°

∠AOC + ∠COD + ∠BOD = 180° (adj. ∠s on a st. line)
44° + 66° + ∠BOD = 180°
∠BOD = 70°

(b)
In ΔOBC :
OB = OC (radii of same circle
Hence, ∠OCB = ∠OBC
∠OCB = 22°

In ΔOEC :
∠CED = ∠OCB + ∠COD (ext. ∠ of Δ)
∠CED = 22° + 66°
∠CED = 88°