Taylor M. answered • 07/28/25
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Passionate Math & Science Tutor | Spelman College STEM Student
By the intersecting chords theorem (Power of a Point):
AP×PC=BP×PD. Since AC=6 and CP=1 then:
AP=6−1=5 Therefore,
BP×PD=AP×PC=5×1=5 Answer:
5
Doug C.
Angle BAC and Angle BDC are inscribed angles intercepting the same arc (BC). Seems those angles must be congruent?
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07/29/25
Christina P.
tutor
But as to the question asked, Taylor's answer is correct, and I'm not sure what the angels have to do with anything.
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07/29/25
Christina P.
tutor
Mark, to answer your question, keep in mind that we don't know what kind of a quadrilateral this is. There is a diameter that come from A to an unknown point, and B and C must be equidistant from that point in order for AB to equal AC.
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07/29/25
Christina P.
tutor
I draw a picture of how it might look, and included the center a the unknown point that would form the diameter with A.
https://www.desmos.com/geometry/nojitu1yk4
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07/29/25
Mark M.
Christina P.: The quadrilateral is cyclic, that is all corners lie on a circle. A and C by standard convention are on opposite vertices of the quadrilateral. I suggest you draw a diagram.
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07/29/25
Doug C.
It is possible for diagonal AC to have the same length as side AB of the cyclic quadrilateral ABCD. Consider isosceles triangle ABC with A as its vertex. AC = AB. Now consider the circumcircle for that triangle. Pick any point D on that circle located on arc AC. The quadrilateral is cyclic and AB still equals AC. desmos.com/calculator/aplbowns2v
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07/30/25
Mark M.
Doug C. Thank you for the demonstration.
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07/31/25
Mark M.
AB is a side. AC is a diagonal. How can AB = AC?07/28/25