Question

如圖24,已知ABCD是矩形紙片,E是AB上一點(diǎn),BE∶EA =5∶3,EC=,把△BCE沿折痕EC翻折,若B點(diǎn)恰好落在AD邊上,設(shè)這個(gè)點(diǎn)為F,

圖24

(1)求AB、BC的長(zhǎng)度各是多少;

(2)若⊙O內(nèi)切于以F、E、B、C為頂點(diǎn)的四邊形,求⊙O的面積.

Answer 1

思路分析:考察所給的條件,翻折△BCE,則△CBE≌△CFE,這樣圖形中提供了很多的線段相等、角相等.

解:(1)連結(jié)CE、CF、EF,設(shè)BE =5x,EA =3x.?

∵四邊形ABCD是矩形,?

∴AB =CD =8x,AD =BC,∠B =∠A =∠D =90°.?

∵△CBE≌△CFE,?

∴EF =5x,FC=BC,?∠CFE =90°.??

∵∠AEF +∠EFC+∠DFC=180°,?

∴∠AFE +∠DFC=90°.?

又∵∠AEF +∠AFE =90°,∠AEF =∠DFC,?

∴sin∠AEF =sin∠DFC,即=.?

∴=,則FC =10x.?

∴ ==.?

∴x =3.∴AB =24,BC =30.?

(2)∵CE平分∠FCB和∠FEB,∴O在EC上.?

設(shè)⊙O和BC切于M,和AB切于N,連結(jié)OM、ON,設(shè)⊙O的半徑為r,?

∴OM⊥BC,ON⊥AB.∴OM∥AB,ON∥BC.?

∴OM =BN =ON =BM =r.?

∴=,即=.∴r =10.?

∴⊙O的面積為100π.