A tangent is a straight line that touches the circumference of a circle at only one point. The angle between a tangent and the radius is 90˚.
Work out the size of each angle marked with a letter. Give reasons for your answers.
Problem 1 :
Solution :
Given, ∠APB = 68˚
∠APB + ∠PBO + ∠AOB + ∠PAO = 360˚
(angle sum property of quadrilateral)
∠AOB = 360˚ - (68˚ + 90˚ + 90˚)
∠AOB = 360˚ - 248
∠AOB = 112˚
Problem 2 :
Solution :
∠OAP = 90˚
∠OAB + ∠BAP = 90˚ ---> (1)
In triangle OAB,
∠OBA + ∠OAB + ∠AOB = 180˚
Let ∠OBA = x
Since OA = OB, ∠OBA = ∠OBA
x + 132˚ + x = 180˚
2x + 132˚ = 180˚
2x = 180˚ - 132˚
2x = 48
x = 48/2
x = 24
By applying x = 24 in (1), we get
24 + ∠BAP = 90˚
∠BAP = 90 - 24
b = ∠BAP = 66˚
Problem 3 :
Solution :
∠OAB = 27˚, ∠OBA = 27˚
Finding c :
∠OAB + ∠OBA + ∠AOB = 180˚
27 + 27 + c = 180˚
154 + c = 180
c = 180 - 154
c = 126˚
Finding d :
∠OBA + ∠ABP = 90
27 + d = 90
d = 90 - 27
d = 63
Problem 4 :
Solution :
∠OBP = 90˚
∠OBA + ∠ABP = 90˚
e + 46 = 90
e = 90 - 46
e = 44˚
∠OBA = ∠OAB = 44˚
∠OBA + ∠OAB + ∠AOB = 180˚
44 + 44 + f = 180˚
88 + f = 180
f = 180 - 88
f = 92˚
Problem 5 :
Solution :
∠PBA = ∠PAB = g
∠BPA + ∠PBA + ∠PAB = 180˚
56 + g + g = 180
2g = 180 - 56
2g = 124
g = 62˚
Since OB is the line drawn from center of the circle to the point of contact of the tangent
g + h = 90
62 + h = 90
h = 90 - 62
h = 28˚
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