Question

theta + beta = ?

a)" "75

b)" "90

c)" "None

Solution

We need to solve this question in two parts.

In the first figure of a circle we can find that the angle of the traingle at tangent point is 90deg as diameter makes an angle 90deg on the circle.

Now using straight line concepts we know sum of angle of a straight line is 180deg

Hence, theta + 90 + 30 = 180

theta = 180 - 120

=> theta = 60

In the second part we can solve using tangent property of the circle.

BQ = BP, AP = AR and QC = CR

Also using pythagoras theorem we can get value of AC = sqrt(AB^2 + BC^2)

=> AC = sqrt(8^2 + 6^2)

=> AC = 10

Now Suppose AR = x

Hence AP = x, RC = (10-x) , QC = (10-x), BQ = (x-4) , PB = (x-4)

Hence, x + (x-4) = 8

=> 2x = 12

=> x = 6

Then, BQ = r = (x-4) = (6-4) = 2

It is given that r = cosecbeta

Therefore, cosecbeta = 2

=> sinbeta = 1/2

hence, beta = 30deg

So, theta + beta = 60deg + 30 deg = 90deg!!