Misleading circles

Question

The picture is drawn to scale.

The smaller circle, radius 12 units, has centre O, and C and D are where the circles intersect. OB is 5 units in length. Point A, the centre of the larger circle, is colinear with points O and B. Find the length of BC, accurate to 3 decimal places.

WARNING: This question is not as straightforward as it looks!

Answer 1

Join AC

AC = 17 (diameter of the big circle)

Let angle OAC = x,

AC = o, OA = c, OC = a

In triangle OCA by using cosine law we get:

cos(x) = (o^2 + c^2 - a^2)/(2*c*o)

=> cos(x) = (17^2 + 12^2 - 12^2)/(2*12*17)

=>cos(x) = 17/24

=> x = 44.900527961

Now angle BOC = 2*x  ( because of triangle exterior angle theorem)

Therefore angle BOC = 89.801055922

Now let BC = m, OB=n and OC = b and angle BOC = y

Using cosine law in triangle BOC we get

m^2 = n^2 + b^2 - 2*n*b*cos(y)

=> m^2 = 5^2 + 12^2 - 2*5*12*cos(y)

=> m^2 = 25 + 144 - 120*0.0034722222

=> m = 12.9839644691

Therefore BC = 12.984 units

Comments

Edit: line 2

AC = 17 (radius of the big circle)

---

Correct! Well done!

---

Thank You Sir,

You forgot my reward.