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Finding external Circumference of 2 overlapping circles

I marked the top of the intersection (where the 2 circles meet) "a" and the bottom "b". Then I made a triangle with the midpoint of AB and the centre of the larger circle. I used Sine Law to find the angles and then found the sector length. My final answer came to be 98.89. However my textbook has a different answer, am I doing something wrong here.

Re: Finding external Circumference of 2 overlapping circles

If you make isosceles triangles using the radii and the length AB you should have a 12, 12, 15 triangle, and a 9, 9. 15 triangle.

Since in each case you have three sides and an unknown angle, I would use the cosine rule to find the unknown angle.

Triangle 1:

So the length of the major segment for Triangle 1 is .

Now do the same for the second circle and triangle.

Re: Finding external Circumference of 2 overlapping circles

Re: Finding external Circumference of 2 overlapping circles

The textbook's answer is 102 cm, but I believe it may have just been a matter of rounding as my answer and Soroban's are very close to the textbook's.