Answer:
Explanation:
You want to know the value of x and the corresponding segment lengths where secant LJ has segment lengths LK=(x-1) and KJ=(x+11), and tangent LM is length 12.
Secant tangent relation
The product of the segment lengths from the intersection point of the secant and tangent to the intersection points of the secant with the circle is the square of the tangent length.
LM² = LK·LJ
12² = (x -1)(x -1 +x +11) . . . . . fill in given expressions
2x² +8x -154 = 0 . . . . . . . put in standard form
x² +4x -77 = 0 . . . . . . . divide by 2
(x -7)(x +11) = 0 . . . . . factor
x = 7 . . . . . . . . . . . the negative solution is extraneous
Segment lengths:
LK = 7 -1 = 6
KJ = 7 + 11 = 18
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Additional comment
When two secants meet at a point outside the circle, the product of lengths to the near and far circle intersection points is the same for both secants.
You can consider a tangent to be a secant where those two intersection points are the same point. Hence the product is the length of the tangent multiplied by itself.
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