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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: (external)(whole) are equal for both secants.
For the secant through C and D: external = CE = x+4, whole = CE+CD = (x+4)+1 = x+5.
For the secant through A and B: external = BE = x+1, whole = BE+BA = (x+1)+11 = x+12.
Set (x+4)(x+5) = (x+1)(x+12):
x^2 + 9x + 20 = x^2 + 13x + 12 → 9x+20 = 13x+12 → 8 = 4x → x = 2.
Answer: 2
For the secant through C and D: external = CE = x+4, whole = CE+CD = (x+4)+1 = x+5.
For the secant through A and B: external = BE = x+1, whole = BE+BA = (x+1)+11 = x+12.
Set (x+4)(x+5) = (x+1)(x+12):
x^2 + 9x + 20 = x^2 + 13x + 12 → 9x+20 = 13x+12 → 8 = 4x → x = 2.
Answer: 2
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