Asked by john
A circle is shown. Secants D E and A E intersect at point E outside of the circle. Secant D E intersects the circle at point C and secant A E intersects the circle at point B. The length of D C is 1, the length of C E is x + 4, the length of A B is 11, and the length of B E is x + 1.
What is the value of x?
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All Answers 1
Answered by
GPT-5 mini
AI
Use the secant-secant power theorem: for external point E,
EC · ED = EB · EA.
Order along each secant is E–C–D and E–B–A, so
ED = EC + CD = (x+4) + 1 = x+5,
EA = EB + BA = (x+1) + 11 = x+12.
Set up equation:
(x+4)(x+5) = (x+1)(x+12)
x^2 + 9x + 20 = x^2 + 13x + 12
−4x + 8 = 0 → x = 2.
Answer: 2
EC · ED = EB · EA.
Order along each secant is E–C–D and E–B–A, so
ED = EC + CD = (x+4) + 1 = x+5,
EA = EB + BA = (x+1) + 11 = x+12.
Set up equation:
(x+4)(x+5) = (x+1)(x+12)
x^2 + 9x + 20 = x^2 + 13x + 12
−4x + 8 = 0 → x = 2.
Answer: 2
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