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邹日僑翠
二次星
日四十月四年〇七九一鹰公年九十五國民華中育教僑華
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Example 2: In rs. AABC, LC - rt., BC // AC,
ED - 243. Prove that DAC 1/3/BAC
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Proof Let be the mid-pt. of ED
Join BI In rt. ABE, N is the mid-pt. of hypotenuse DE
..NBND NE
ND
ABAN
BA - BN (LED) Proved * n (base ■ isos, A
J. END (proved)
3 - 4 (base La 1808
1,770英文中學會考試題預習專欄
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數學科
廿四)
MATHEMATICS (24).
equalities Triangles (Including the 5 centres.
Important Theorems for referenos
For con ruence of triangles t
two sides and an included angle (SAS)
two angles and a side (13A or AAS*
three sides (SS$)
a right angle, a hypotenuse and a side (RHS)
For isosceles triangles:
1. bases, isos. ▲ (are equal)
2. sides opp. equal ́s (are equal)
3. bisector of vert. bisects also the base
at t. (The converse is also true)
4. bisector of the ext. of the vert.
parallel to the base.
0) For right-angled triangle t
5. It is always equidistant from the mid-point of the hypotenuse to the three vertices,
6. Hypotenuse is the greatest side in a rt. 7. If one acute / equals 60° (or 30°), the hypotenuse is twice the shortest side, For any triangle (on inequalities) 8. Ext. / of a ▲
..
any int. opp.
9. Any two sides together the third side 10. The difference between any two sides
third side.
11, The greater side has the greater opf. La
(The converse is also true)
The five centress
12, The - bisectors of the three sides of a ▲
are concurrent. (Circumcentre throrem). The circumcentre is equidistant from the three vertices.
13. The bisectors of the int, a of a A are
concurrent. (In-centre theorem). The in- bentre is equidistant from the three sides, The bisector of one int, and two other ext. biaactors are concurrent. (Ex-centre theorem), The ex-centre is equidistant from the three sides,
15. The thres medians of a A are concurrent.
(Centroid theorem). The controid is 1/3 of the way along each median measured towards the vertex.
NOTES:
The thres altitudes of a (Orthocentra theorem).
Aare concurrent
a) The ▲ whose vertiose are the fast of
the altitudes is called the PEDAL TRIANGLE of the original.
(b) If the ▲ is acute-angled, the
orthocentre lies inside the ▲ I the ortho centra lies outside the ▲ for obtuse-angled and lies on the vertex of the rt. for art.
Orthocentre is NOT the centre of any important circle assoolate with the
"The word "centre" is often used zo. denote the common point of intersection of three or more conourrent lines,
If A is equilateral, then, the oircumcentre, incentre, orthocentre and centroid all coincido with each other.
2. Conversely, if any two of the circumcentre; in-
centre, orthocentre and centroid coincide one another, then the triangle is equilateral.
EXAMPLES
Example 1's In cu.ẩy hưu, m 18 the mid-point of tha hypotenuse BC. Draw XDBC such that D lies on opposite side of A and that DK 1BC.
Frove that AD bisects |__BAG,
Givens ▲ ABC with LA - 1 rt./
DM - MO - MD
MDL BO
To prove : AD bisecte / BAC
Proofs Join AM and draw AN // MD
MD 1BC
ANALEC
Since M is the mid-point of BC.
TH
KADD (bases ison. A
And, LD - / DAN (alt. £9, MD // AN)
|_ MAD = /__DAN Since XA XE (Prove)
(1)
MABB complement of C
In rt,▲ ANGI / NAC - complement of
(1) + (2)
MAB NAC
/ BAD
CAD
• AD bisecte Z BAO
(2)
- 2 (ext, at A)
(alt. / s BD //AC)
- } |_ BAC
Example 3: In ▲ABC, „B>AC, take pointe D E on
1B, AC produced resp. such that AD AE
(AB AC). Prove that DE bisecta BC.
Given: ABC with AB > AC
AD AF 1(AB + 40)
To prove
DE bisects. BC Proofs Let DS outs BC at. F
Draw CF //AB and
Lat CF out DE at F.:
ADAE (AB+ AC)
AB + AC - AD + ÁF
ox ABAD - AB - AC
BD. Os
AD - AE and BD - CF (given & consin..
UF UN
LE (= LADE) (COTT./ ® AT//CF
baze Zaj tile A (Sides opp. equal (7)
DBFC is « //gram (304 OP)
BP - PC (diag. //zram) DE bisecta BC
Example 4: In AABC, AB > AC, BC, CE are the
bisectors of B, /C rasp.; prove BD >GE.
Proof: The proof is givon foro in
outline form.
(1) L_ACB-> L_ABD
(2) Draw CX-
such that
ECX-
Then / BCX >__ CBX BX > CX
(3) Take on BX e.t. PM + CX (N 119s between B
and I). Draw MN // CZ (N 110s tsineso R. and D) Then ▲CEX ABMN (ASA)
CE BN
(4)
3D and BEN CE (BD > CE)
Example 5: Prove that the distance of the orthocentri
of a triangle from a vertex is twice, the distance of the circumsantza from the opposite side..
Froof: Let 0, be the circumcontre
and orthocentre of AABC
resp. and let
mid-pt. of
Join My Po be the
ABG
He resp.
BQ = QA (Given)
BP PC (Given
Pq 130 (Mid-point theorem)
Similarly, in AREC, MN AC
PQMN
OQLAB end CELAB (on OPL BC and AÐ.
D. BO (OP
the aides of
OPQ, HMN are
each other the ints are correspondingly equal
PQ- MN proved.
AOPQ = AFMN (ASA)
OP = HM (HA)
HA 201
Similarly, HO- 200
Or, the problem nav alas be proved as ToLLOWE : With centre 0, radius OB, draw the oirounoircis of ABC. Produce BO to meet the circumeirole at I. Then, (1) AECX 10 a//gram 8.t. CX - AH
(11) ABCX is rt. at C and OP – cz
Example 6: If AD, BE, CF are medians of AABC, prove
that AD + BE + CF (AB + 80 + CA)
Proof: Let @ be the centroid of AABC From AG BC1481GC
1.0* BE+CF >BC Similarly, by the sum of 2 sides.
In AGACS CF + AD > CA
In AGAD, ·AD +2°
中國麻中心預習題答案
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文臣,太宗及休,亦能 學軍大敗。
寄科取士,多
自密以後,已有三教合物之說,若規打期天確,已趨會為一,乃至生理學。 若於上述因素,未爆於得以大盛怨束。
仟二)何謂經學大學限?其代表人物您能望並純染大風至宝機。
廿三
(二)際大學派的代签人物—源族派因是RR;學派组腐、使用爲代法;開
(三)四大學派的重要主張————卷學派的所收,太極說,主張事,惟饿問無欲 ,無欲珝明心,明心見法,為化地之山字。塗明派的蔬類、糧頭兄則 全反識仁佛本,而以誠敬育之。又涵義製版,你不在該海關繫藏主張以 易宗,以中將您體,以孔孟馬法。艾常以「登天地立心,您生民立命,你往類滋 學,您館世開太平!你可致人之本。世德羅得先坐。開學派的朱務,其破源於二營 *為宋代理學之集大成者。張所以窮無受其州,反網以路其實,球心㬅敬够主
●故無常以「搭政部用發入治學的主要功夫。他的回答保註,影前後版大。 和隆象山的學說有何不同 答:朱凞和館象山(九日是海末约理明大家。他們都是雷心性,都只能做人之潭,但他們 的立證,邮並不相同。二人有在江湖,舉行,使彼此沒法說服對方,米 以「嬸略年確」,陳期以米然得了支难殄」,总能有所謂「未異同」 朱所的 學快組優,問重於外句的「铭为理工,與物物,若一架湫碎塊,用 力飲久,自然會豁然貫法。但象山之學制電內。認愛「心理上說,確不自事物探 果,只要渡其本心便可。故仙帶:搖物只思的心,商做學問的工夫,只是去此已的物欲之 發,即恢復其本競,所以修萬物權於心,不錄於向外求。可見你氏之學,新電向外求 TUTE BABAEZA TRAK HAZIRAZIOA EZTE, S (管理)宋代文障及史學的發酒絡之
答:(一)宋代的文學
宋代的文學,政制及古文魚素,調事於野中,茶抗代時始拋出,两家而大成。 於宋村司之學 代,有問請用來,故澄清,或換句子要回不一,
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•WKER, MARKET'S PENJ BIEN-
·郭響不少,如不際,跟然,大抵宋初詞的形式
早期的開步以要說夠夠器材,但自家索班及中放入同,洞鄆又一大 一
然不能,只和族有之,成噹放之宗師。至於古玄亦盛,自歌陽永叔漫
SAZNA, 523 SHKAK • ZKUSSE · EZT • Mi • A
1321? BEZES ▼ BRISBEFAXX?
(二)宋代的史學
宋代的史學發達,以前馬光的資給警號碼及響。以編年嬬體,上起戰國,下次开代 饷盒一部枵鉅餐。此外尚有飛居工等的五代史,鼠器堡的新海去,同解無史,之外又有
的通志、寫難的夕肤需老和梵做的酒典合生活。都是史學上的鉅著。
光頭
̇中國歷史科預習問題
(六)元朝(公元1277年一一公元1368年)
自(A)又說一體,傳奇(B)爲朗太限朱元飛所钱,共享國八十九年的世 古前三次西征:第一次是讲成吉思汗帶掛;第二式由(A);第淡(B
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4.中國三大帶明於何時傳至歐洲?
CADR
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(四)蒙古入主中原後,對其寶島采取同饮浓 (五)熙略淞元朝华海交通之大略带形,及其 (六)元朝的時候,中國有那些文物傳照!跟生
BE > AB
Adding,
(AD + B5 + CF) AB + BO
CA
AD
BE
CA)
X, Y, Z are the mid-pt. of the sides or
ABC; prove that the orthocentre of XYZ ie the oircumcentre of AABC.
of: Lat O be the ortnocentre
AITZ
Join Xu and produce
Put YZ at P.
XP YZ
2 are mid-pt. of AC, "BC"
TZ // AB (mid-
that
IPULIZ" and "YZ. // AB
PIL AB
108 X is the
mid-pt. of AB (Given)
*** PX bisects AB of rt./i 1.e. OX bisects AB at rt.s Similarly, Or bisecta AC at rt./ » Hence, 0 is also the circumcentre of
Example
ABC.