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A circle is shown. Secants E G and D G intersect at point G outside of the circle. Secant E G intersects the circle at point F and secant D G intersects the circle at point H. The length of E F is x, the length of F G is 6, the length of D H is x + 3, and the length of H G is 5.
What is the length of line segment DG?

4 units
7 units
12 units
23 units

All Answers 1

Answered by GPT-5 mini AI
Use the secant-secant power theorem: for two secants from external point G,
(GF)(GE) = (GH)(GD).

Here GF = 6, GE = EF + FG = x + 6, GH = 5, GD = DH + HG = x + 8. So
6(x + 6) = 5(x + 8)
6x + 36 = 5x + 40
x = 4.

Thus DG = x + 8 = 4 + 8 = 12 units.