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Answer:
3. x = 10
4. a = 8
Explanation:
A reasonably simple rule to remember for secants outside the circle or chords inside the circle: the product of the lengths from the point of intersection of the lines to the points of intersection of the circle is the same for each line.
When the external line is tangent to the circle, the two points of intersection of the line with the circle are the same: the point of tangency. Effectively, this means the length is squared.
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3. For line QP, the product is 3(3+5) = 24.
For line QR, the product is 2(2+x). The rule above says these are equal:
24 = 2(2+x)
12 = 2+x . . . . . divide by 2
10 = x . . . . . . . subtract 2
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4. For line BC, the product is (a)(a) = a^2.
For line GC, the product is 4(12+4) = 64. The rule above says these are equal:
a^2 = 64
a = √64 . . . . take the square root
a = 8