From circle 1 to circle 8 ... learning the hula hoop ?
Referring to the attached pic:
APQ is a semi-circle with PQ as the diameter.
QP is extended to B, and BA is a tangent to the circle at A.
AC is perpendicular to PQ and PC:CQ = 1:4.
AB = 8cm
(a) Prove
(b) Prove
(c) If BP = y and PC = k, prove that
(d) Find BP
(e) Find the radius of circle.
My work:
(a)
triangles APC ~ ACQ (same angles)
hence AC:PC = QC:AC
(b)
triangles BAQ ~ BPA (same angles)
hence AB:BQ = BP:AB
(c)
since PC:CQ = 1:4,
CQ = 4k and hence PQ = 5k
hence
hence we have AC = 2k
(Pythagarus)
AP = (k)(5^0.5)
Also
AQ = (2k)(5^0.5)
y:8 = AP:AQ = k(5^0.5):2k(5^0.5)
y:8 = 1:2
hence y = 4
and 8:BQ = AP:AQ = 1:2
8:(5k + y) = 1:2
8:(5k + 4) = 1:2
16 = 5k + 4
k =
Therefore,
Confused here: relation between k and y is supposed to be proved without working out their values in advance. Is there another method to prove that k = 3y/5 ????
(d)
BP = y = 4cm
(e)
PQ = 5k = 12cm