These involve the rules for the power of a point. For questions 6-8, we use the theorem that the square of the length of the tangent is equal to the length of the secant multiplied by the length of its external segment.
6. 6^2 = (3)(x + 3)12 = x + 3x = 9 units
7. 4^2 = (2)(x + 2)8 = x + 2x = 6 units
8. 24^2 = (12)(x + 12)48 = x + 12x = 36 units
9. The included angle is half the difference of the larger and smaller arcs. Since the larger arc is 85 degrees, the smaller is 25 degrees, and the difference is 60. Half this difference is x = 30 degrees.
10. The angles at the intersection are vertical angles, so both equal to 65 degrees. Then the sum of the intercepted arcs must be equal to double of the vertical angle: 2 x 65 = 130 degrees. Since one is 95 degrees, the other is x = 130 - 95 = 35 degrees.