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# Geometry Question

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1) A, B, C, and D are points on a circle, and segments AC and BD intersect at P, such that AP = 8, PC = 1, and BD = 6. Find BP, given that BP < DP.

2) In 4ABC, AB = 3 and BC = 5. Find the number of possible integer values of CA. [Disclaimer]: In this class, we do not consider degenerate triangles (i.e. a triangle whose area is 0).

3) In obtuse 4ABC, AB = 3 and BC = 5. Find the number of possible integer values of CA

Mar 23, 2020

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1) A, B, C, and D are points on a circle, and segments AC and BD intersect at P, such that AP = 8, PC = 1, and BD = 6. Find BP, given that BP < DP.

Let AC be a side of an inscribed square. Let O be a center of a circle, and N is the intersection point of BD and CO.

AP = 8       PC = 1        BD = 6        ∠(CPN) = 45°

PN = sqrt [(PC)² /2]  =  sqrt(1/2) [ sqrt(50)] /10  =  sin(45°) = cos(45°) = 0.707106781

BP = ( BD/2 ) - PN = 2.292893219 units  Mar 23, 2020
edited by Dragan  Mar 23, 2020
edited by Dragan  Mar 23, 2020
edited by Dragan  Mar 24, 2020
edited by Dragan  Mar 24, 2020