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Consider an isosceles triangle A B C with sides B C=30, C A=A B=20 . Let D be the foot of the perpendicular from A to B C , and let M be the midpoint of A D . Let P Q be a chord of the circumcircle of triangle A B C , such that M lies on P Q is parallel to B C . The length of P Q is:

Correct answer

25

Step-by-step solution

Eq. of P Q: y= 5 7 2 Perpendicular bisector of A C : aligned & (y- 5 7 2 )= -1 ( 5 7 -0 0-15 ) (x- 15 2 ) & (y- 5 7 2 )= 15 5 7 (x- 15 2 ) & Centre intersection of perpendicular bisector & (0, -5 7 ) (0, -5 7 7 ) & Eq ^ n of circumcircle aligned aligned & (x-0)^2+ (y+ 5 7 7 )^2= (5 7 + 5 7 7 )^2 & x^2+ (y+ 5 7 )^2=25 7 64 49 = 25 64 7 aligned Intersection with P Q: y= 5 7 2 aligned & x^2= 25 64 7 - ( 5 7 2 + 5 7 )^2= 25 64 7 - 1 4 7 (45^2 ) & x^2= 25 64 4-45^2 28 = 5^2 2^2 ( 64 4-81 7 )= 5^2 2^2 25 & x= 25 2 distan

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