【答案】
分析:●觀察計算:分別代入計算即可得出

與

的大小關(guān)系;
●探究證明:
(1)由于OC是直徑AB的一半,則OC易得.通過證明△ACD∽△CBD,可求CD;
(2)分a=b,a≠b討論可得出

與

的大小關(guān)系;
●實踐應(yīng)用:通過前面的結(jié)論長方形為正方形時,周長最小.
解答:
解:●觀察計算:

>

,

=

.(2分)
●探究證明:
(1)∵AB=AD+BD=2OC,
∴

(3分)
∵AB為⊙O直徑,
∴∠ACB=90°.
∵∠A+∠ACD=90°,∠ACD+∠BCD=90°,
∴∠A=∠BCD.
∴△ACD∽△CBD.(4分)
∴

.
即CD
2=AD•BD=ab,
∴

.(5分)
(2)當(dāng)a=b時,OC=CD,

=

;
a≠b時,OC>CD,

>

.(6分)
●結(jié)論歸納:


.(7分)
●實踐應(yīng)用
設(shè)長方形一邊長為x米,則另一邊長為

米,設(shè)鏡框周長為l米,則

≥

.(9分)
當(dāng)

,即x=1(米)時,鏡框周長最�。�
此時四邊形為正方形時,周長最小為4米.(10分)
點評:本題綜合考查了幾何不等式,相似三角形的判定與性質(zhì),通過計算和證明得出結(jié)論:


是解題的關(guān)鍵.