F5 Maths - Ans.fyi 參考答案

F5 Maths

2010-10-13 11:50 pm

Figure: http://img339.imageshack.us/img339/5480/72335643.jpg

In the figure, O is the centre of the circle, K is the mid-point of chord AB,
CD is any chord passing through K. Tangent at C and D meet AB produced
at P and Q respectively.

a) Prove that ODQK is a cyclic quadrilateral.

b) Prove that OKCP is a cyclic quadrilateral.

c) Prove that triangle OCP and triangle ODQ are congruent.

回答 (1)

用戶2053

2010-10-17 8:49 pm

✔ 最佳答案

圖片參考：http://img339.imageshack.us/img339/5480/72335643.jpg

a)

ㄥOKQ = 90° (line from centre⊥chord bisects chord)
ㄥODQ = 90° (tangent⊥radius)

ㄥOKQ + ㄥODQ = 90 + 90 = 180°
ODQK is a cyclic quadrilateral (opp. ∠s supp.)

b)

ㄥOCP = 90°(tangent⊥radius)
ㄥOKP = 90°(line from centre⊥chord bisects chord)

OKCP is a cyclic quadrilateral (∠s in the same segment)

c)

ㄥOCP = ㄥODQ = 90° (proved)

ㄥCPO
= ㄥDKO(ext. ∠s, cyclic quad.) = ㄥDQO(∠s in the same segment)

△OCP ~ △ODQ (A.A.)

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