急!急! F4 Properties Circles q5

10.
In ΔCAT :
∠BAC = ∠ACT + ∠ATC (ext. ∠ of Δ)
∠BAC = 12° + 33°
∠BAC = 45°

In ΔABC :
∠BAC + ∠ACB + ∠ABC = 180° (∠ sum of Δ)
45° + 75° + ∠ABC = 180°
∠ABC = 60°

The length of arc and the angle subtends at circumference are in ratio.
arc BC : arc AC
= ∠BAC : ∠ABC
= 45° : 60°
= 3 : 4


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11.
(a)
arc SQ = 3 (arc PR)
arc SQ : arc PR = 3 : 1

The length of arc and the angle subtends at circumference are in ratio.
∠PSR : ∠SPQ
= arc PR : arc SQ
= 1 : 3

(b)
∠PSR : ∠SPQ = 1 : 3
Let ∠PSR = n°, then ∠SPQ = 3n°

∠SXQ + ∠RXQ = 180° (adj. ∠s on a st. line)
124° + ∠RXQ = 180°
∠RXQ = 56°

In ΔPSX :
∠RXQ =∠PSR + ∠SPQ (ext. ∠ of Δ)
56° = n° + 3n°
n = 14
3n = 42

∠SPQ = 42°


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13.
Let OC = y cm
Then radius OD = y + 6 cm

AC + BC = 24 cm
But AC = BC
Then AC = 12 cm

C is the mid-point of AB, then ∠ACO = 90°
OA² = OC² + AC²
(y + 6)² = y² + 12²
y² + 12y + 36 = y² + 144
12y = 108
y = 9

OC = 9 cm