Geometry Question

Question

In the diagram, AP is secant to the circle at points A and B, and CP is secant to the circle at points C and D. The lines intersect outside of the circle at point P.

What is the length of CD?

CD= (x+6)

DP= 7

BP= 8

AB= 13

Answers:

11

22

17

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Answer 1

Known Geometry Principle (it can be proved that the following is true from basic geometry principles):

When two secants of a circle, intersect outside of that circle, the product of the lengths of one of the secants and its external segment equals the product of the lengths of the other secant and its external segment.

Thus, for the problem above this becomes: AP x BP = CP x DP

we are given that: DP = 7 and BP = 8

And, from the description of the figure: AP = AB + BP = 13 + 8 = 21

so, now we can solve for CP (which is not given) in order to find CD, since CD = AP - DP:

from AB x BP = CP x DP

divide both sides by DP to solve for CP:

so CP = (AB x BP)/DP = (21 x 8) / 7 = 24

and CD = CP - DP = 24 - 7 = 17

so the answer is 17! (I encourage you to draw your own picture while doing this).