Answer:
4. (d)
5. (b)
Explanation:
You want to apply the secant and chord rules to the circles given in the diagram.
The secant rule is "the product of distances from the point of intersection of secants to the two points of intersection of the secant with the circle is the same for both secants." A tangent can be considered to be a secant that has both points of intersection with the circle in the same place.
The chord rule is similar to the secant rule: "The product of distances from the point of intersection of the chords to their intersection with the circle is the same for both chords."
4.
The diagram shows a tangent of length 12, so the product of distances to the two points of intersection with the circle is 12(12).
The secant has length 8 to the near point of intersection, and (x+8) to the far point of intersection. That product is the same as for the tangent:
8(x +8) = 12(12) . . . . . . matches choice D
5.
One chord has lengths of 14 and x, so the product of those is 14x.
The other chord has lengths of 12 and 21, so the product of those is 12(21).
The product is the same for both chords, meaning ...
14x = 12(21) . . . . . . matches choice B
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Additional comment
We have purposely tried to write the rules in "parallel" fashion, because we find them easier to remember that way. Considering a tangent to be a secant with the points of intersection with the circle merged means the same rule can be used for all three cases:
- intersecting secants,
- a tangent intersecting a secant, and
- intersecting chords.