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circle

[匿名]

2012-07-19 2:03 am

Refer to the figure:
http://imageshack.us/photo/my-images/217/1001la.jpg/
TB and TD are tangents to the circle at B and D respectively. PQ and CD are parallel chords of the circle with centre O. BKC and PKQT are straight lines. If M is the mid-point of OT and angle BCD = 70 degree, fine angle TMD.

更新1:

Find angle TMD Provided that: 1. T, B, O and D are concyclic 2. T, B, K and D are concyclic

回答 (1)

用戶10012

2012-07-19 11:33 pm

✔ 最佳答案

It is tedious to answer step by step, here giving you the keys:
(1)Since OD is perpendicular to DT(property of tangent to circle) and OT is a straight line, so OBTD is a circle with OT as diameter and M is the center of circle. So OM = MT = MD are radius of circle. Triangle ODM is an isos. triangle with angle MOD = angle MDO.
(2) Angle BOD = 2 x angle BCD = 2 x 70 = 140 degree (angle at center 2 times angle at circumference.) But angle BOD = 2 x angle MOD (triangle TBO congruent triangle TDO). So angle MOD = 70 degree
So angle TMD = 2 x angle MOD = 140 degree (ext. angle of triangle.)
Note : Answer is regardless of TBKD concyclic or not. TBOD is certainly concyclic

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