A LINE DRAWN FROM CENTER OF CIRCLE TO THE TANGENT LINE

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˚

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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