第二十五章 锐角的三角比·拔尖卷
Ⅰ
!"#$ 10 '()* 30 *(,' 3 *-
1 #3 *-#24-25 ./01·2345·6789-:;(<=>𝐴𝐵𝐶𝐷?(@ E <𝐷𝐶1(A=>B𝐴𝐸CD(
E@ D F<𝐵𝐶G1H@ F I J𝐴𝐵= 3(𝐵𝐶= 5(Ktan∠𝐷𝐴𝐸HLM# -
A
1
2
B
9
20
C
2
5
D
1
3
D
* NOPQR=>HSTUCDHSTUVWXYZ[\]^_`ab(c8defabg
h ijkl=>HSTZ[CDHSTmX𝐴𝐹= 𝐴𝐷= 5,𝐷𝐸= 𝐹𝐸(n<Rt △𝐴𝐵𝐹?(oVWX
YmX𝐵𝐹HL(pqrs𝐶𝐹= 𝐵𝐶−𝐵𝐹= 1tu𝐷𝐸= 𝐹𝐸= 𝑥(K𝐶𝐸= 3−𝑥(<Rt △𝐶𝐸𝐹?(oVWXY
mX𝑥HL(v𝐷𝐸=
5
3(wxoyzHX{(vr|s
}
∵~G>𝐴𝐵𝐶𝐷M=>(𝐴𝐵= 3(𝐵𝐶= 5(
∴∠𝐵= ∠𝐶= ∠𝐷= 90°,𝐴𝐵= 𝐶𝐷= 3,𝐵𝐶= 𝐴𝐷= 5(
∵A=>B𝐴𝐸CD(E@ D F<𝐵𝐶G1H@ F I(
∴𝐴𝐹= 𝐴𝐷= 5,𝐷𝐸= 𝐹𝐸(
∴<Rt △𝐴𝐵𝐹?(𝐵𝐹=
𝐴𝐹2−𝐴𝐵2 =
52−32 = 4(
∴𝐶𝐹= 𝐵𝐶−𝐵𝐹= 1(
u𝐷𝐸= 𝐹𝐸= 𝑥(K𝐶𝐸= 𝐶𝐷−𝐷𝐸= 3−𝑥(
<Rt △𝐶𝐸𝐹?(rs𝐶𝐹2 +𝐶𝐸2 = 𝐸𝐹2(v12 + (3−𝑥)2 = 𝑥2(
s𝑥=
5
3(v𝐷𝐸=
5
3(
∴<Rt △𝐴𝐷𝐸?(tan∠𝐷𝐴𝐸=
𝐷𝐸
𝐴𝐷=
5
3
5 =
1
3
!
D
2 #3 *-#2025·235·-:;(
a𝐴𝐷∥𝐵𝐶(𝐵𝐷⊥𝐴𝐶(𝐴𝐶= 4(𝐵𝐷= 8(Kcos∠𝐷𝐵𝐶=
#-
A
5
5
B
2 5
5
C
3 5
5
D
1
2
B
* NQR~G>HXST(VWXY(yHX{(deab@HgH
h 𝐷𝐷𝐸∥𝐴𝐶(𝐵𝐶@𝐸(~G>𝐴𝐷𝐸𝐶g~G>(K𝐴𝐶= 𝐷𝐸= 4(noVWXY
𝐵𝐸= 4 5(wxocos∠𝐷𝐵𝐶=
𝐵𝐷
𝐵𝐸vr
}
𝐷𝐷𝐸∥𝐴𝐶(𝐵𝐶@𝐸(
∵𝐴𝐷∥𝐵𝐶(
∴~G>𝐴𝐷𝐸𝐶g~G>(
∴𝐴𝐶= 𝐷𝐸= 4(
∵𝐵𝐷⊥𝐴𝐶(
∴𝐵𝐷⊥𝐷𝐸(
∴∠𝐵𝐷𝐸= 90°(
∴𝐵𝐸=
𝐵𝐷2 + 𝐷𝐸2 =
82 + 42 = 4 5(
∴cos∠𝐷𝐵𝐶=
𝐵𝐷
𝐵𝐸=
8
4 5 =
2 5
5 (
!
B
3 #3 *-#2025··-<:;H'y> ¡?(A(B(C(D ¢M'y>H£@(
7𝐴𝐵¤𝐶𝐷f@ O(Ksin∠𝐴𝑂𝐶HLM#-
A
1
2
B
2
2
C
3
2
D ¥¦mX
B
* NQR§]\]>(¨©ª ¡«@¤∠𝐷𝐴𝐵= ∠𝐶𝐴𝐸g¬Hh
®¯𝐴𝐷U𝐵𝐷(°¡@𝐸(:;(u'y>HGM 1 (K±y>HSTs²𝐵𝐷=
2,𝐴𝐷= 2 2
,∠𝐴𝐷𝐸= ∠𝐵𝐷𝐸 = 45°(K∠𝐴𝐷𝐵= 90°(nklyzHX{(<Rt △𝐴𝐷𝐵?r³´tan∠𝐷𝐴𝐵=
1
2(<Rt
△𝐶𝐷𝐸?r³´tan∠𝐶𝐴𝐸 =
1
2(Z∠𝐷𝐴𝐵= ∠𝐶𝐷𝐸(wx±\]>µ]ST¤]H¶·r
∠𝐴𝑂𝐶= ∠𝐴𝐷𝐸= 45°(Zsin∠𝐴𝑂𝐶= sin45° =
2
2
}
®¯𝐴𝐷U𝐵𝐷(°¡@𝐸(:;(u'y>HGM 1 (
K𝐵𝐷=
12 + 12 =
2,𝐴𝐷=
22 + 22 = 2 2,∠𝐴𝐷𝐸= ∠𝐵𝐷𝐸 = 45°(
∴∠𝐴𝐷𝐵= ∠𝐴𝐷𝐸+ ∠𝐵𝐷𝐸= 90°(
<Rt △𝐴𝐷𝐵?(tan∠𝐷𝐴𝐵=
𝐷𝐵
𝐴𝐷=
2
2 2 =
1
2(
<Rt △𝐶𝐷𝐸?(tan∠𝐶𝐷𝐸=
𝐶𝐸
𝐷𝐸=
1
2(
∴∠𝐷𝐴𝐵= ∠𝐶𝐷𝐸(
∵∠𝐴𝑂𝐶= ∠𝐷𝐴𝐵+ ∠𝐴𝐷𝐶(
∴∠𝐴𝑂𝐶= ∠𝐶𝐷𝐸+ ∠𝐴𝐷𝐶= ∠𝐴𝐷𝐸= 45°(
∴sin∠𝐴𝑂𝐶= sin45° =
2
2 (
!
B
4 #3 *-#24-25 ./01·¸¹º»·6789-:;(< △𝐴𝐵𝐶?(∠𝐵𝐴𝐶= 90°(𝐴𝐵= 20(𝐴𝐶= 15(
△𝐴𝐵𝐶H¼𝐴𝐷]*𝐶𝐹@ E(K
𝐷𝐸
𝐴𝐹HLM# -
A
3
5
B
3
4
C
1
2
D
2
3
A
* NOPQ\]^_H(HhgdeVWXYU\]^_HX{ ±VWXY
𝐵𝐶H(nkl§]\]>H½¾¿𝐴𝐷H(±VWXY𝐷𝐶H(wx±]*HX
{(rs²tan∠𝐴𝐶𝐹= tan∠𝐸𝐶𝐷(wx±À]\]^_HX{(Ár𝐷𝐸𝐴𝐹HÂL
}
< △𝐴𝐵𝐶?(𝐵𝐶=
𝐴𝐵2 + 𝐴𝐶2 =
202 + 152 = 25(
∵𝐴𝐷g △𝐴𝐵𝐶H¼(
∴
1
2𝐴𝐷⋅𝐵𝐶=
1
2𝐴𝐵⋅𝐴𝐶(
∴25𝐴𝐷= 20 × 15(
∴𝐴𝐷= 12(
<Rt △𝐴𝐷𝐶?(𝐷𝐶=
𝐴𝐶2−𝐴𝐷2 =
152−122 = 9(
∵𝐶𝐹*∠𝐴𝐶𝐵(
∴∠𝐴𝐶𝐹= ∠𝐸𝐶𝐷(
∴tan∠𝐴𝐶𝐹= tan∠𝐸𝐶𝐷(
∴
𝐴𝐹
𝐴𝐶=
𝐷𝐸
𝐷𝐶v
𝐷𝐸
𝐴𝐹=
𝐷𝐶
𝐴𝐶=
9
15 =
3
5(
∴
𝐷𝐸
𝐴𝐹=
3
5
!
A
5 #3 *-#2025·2÷?Ä-:;(CDy>𝐴𝐵𝐶𝐷H G𝐵𝐶(E@𝐶F<𝐵𝐷1H@𝐹I(CÅ𝐵𝐸
𝐴𝐶@𝐺 J𝐷𝐸= 2 2(K𝐶𝐺Hg# -
A 2
B 2
C 2 +1
D 2 2−1
B
* :;(𝐺𝐺𝐻⊥𝐵𝐶𝐻(oÆCrs
𝐵𝐶= 𝐵𝐹(𝐶𝐸= 𝐸𝐹(∠𝐵𝐹𝐸= ∠𝐵𝐶𝐸= 90° = ∠𝐷𝐹𝐸(
∠𝐹𝐵𝐸= ∠𝐶𝐵𝐸(∠𝐷𝐸𝐹= ∠𝐹𝐷𝐸= 45°(q𝐷𝐸= 2 2(rs𝐷𝐹= 𝐸𝐹= 𝐷𝐸⋅sin45° = 2(
𝐶𝐷= 𝐵𝐶= 2 2 +2 = 𝐵𝐹(𝑂𝐵=
1
2𝐵𝐷=
2 +2(𝑂𝐺= 𝐻𝐺(Rt △𝑂𝐵𝐺≌Rt △𝐻𝐵𝐺(rs𝐵𝐻= 𝐵𝑂=
2
+2(np Çvr.
}
:;(𝐺𝐺𝐻⊥𝐵𝐶𝐻(
∵y>𝐴𝐵𝐶𝐷(
∴𝐵𝐶= 𝐶𝐷= 𝐴𝐵= 𝐴𝐷(∠𝐵𝐶𝐷= ∠𝐴𝐷𝐶= 90°(∠𝐷𝐵𝐶= ∠𝐵𝐷𝐶= 45°(𝐴𝐶= 𝐵𝐷(𝑂𝐴= 𝑂𝐶= 𝑂𝐵= 𝑂𝐷(
𝐴𝐶⊥𝐵𝐷(
oÆCrs
𝐵𝐶= 𝐵𝐹(𝐶𝐸= 𝐸𝐹(∠𝐵𝐹𝐸= ∠𝐵𝐶𝐸= 90° = ∠𝐷𝐹𝐸(∠𝐹𝐵𝐸= ∠𝐶𝐵𝐸(
∴∠𝐷𝐸𝐹= ∠𝐹𝐷𝐸= 45°(q𝐷𝐸= 2 2(
∴𝐷𝐹= 𝐸𝐹= 𝐷𝐸⋅sin45° = 2(
∴𝐶𝐷= 𝐵𝐶= 2 2 +2 = 𝐵𝐹(
∴𝐴𝐶= 𝐵𝐷= 𝐵𝐹+ 𝐷𝐹= 2 2 +4(
∴𝑂𝐵=
1
2𝐵𝐷=
2 +2(
∵∠𝐹𝐵𝐸= ∠𝐶𝐵𝐸(𝐺𝐻⊥𝐵𝐶(𝐴𝐶⊥𝐵𝐷(
∴𝑂𝐺= 𝐻𝐺(
∵𝐵𝐺= 𝐵𝐺(
∴Rt △𝑂𝐵𝐺≌Rt △𝐻𝐵𝐺(
∴𝐵𝐻= 𝐵𝑂=
2 +2(
∴𝐶𝐻= 𝐵𝐶−𝐵𝐻=
2(
ÈYrs
𝐶𝐻= 𝐺𝐻=
2(
∴𝐶𝐺=
2 + 2 = 2t
!
B.
@ÉNQHgy>HST(`Ê\]>HXST(]*HST(Ë`\]>HX
ST(À]\]^_H(VWXYH(ÌÍHÎÏgNHh.
6 #3 *-#2025·~ÐÑ»·\-<Rt △𝐴𝐵𝐶?(∠𝐶= 90°(𝐴𝐶= 8(@ D M𝐴𝐵?@(@ D
𝐷𝐸⊥𝐴𝐵(𝐴𝐶@ E(®Ò𝐶𝐷(Ósin∠𝐶𝐷𝐸=
7
25(K𝐷𝐸HÔg# -
A
3 34
5
B
15
4
C 3
D 4
B
* 𝐸𝐸𝐻⊥𝐶𝐷𝐻(osin∠𝐶𝐷𝐸=
𝐸𝐻
𝐷𝐸=
7
25(u𝐸𝐻= 7𝑥(K𝐷𝐸= 25𝑥(∠𝐴= ∠𝐴𝐶𝐷(rs
sin∠𝐴= sin∠𝐴𝐶𝐷(
𝐶𝐸=
7
4(𝐴𝐸= 8−
7
4 =
25
4 (ÈYrs
cos∠𝐴=
𝐴𝐷
𝐴𝐸=
𝐴𝐶
𝐴𝐵(q𝐴𝐵= 2𝐴𝐷(rs
𝐴𝐷= 5(p Çrs¯Õ
}
𝐸𝐸𝐻⊥𝐶𝐷𝐻(
∵sin∠𝐶𝐷𝐸=
𝐸𝐻
𝐷𝐸=
7
25(
∴u𝐸𝐻= 7𝑥(K𝐷𝐸= 25𝑥(
∴𝐷𝐻=
𝐷𝐸2−𝐸𝐻2 = 24𝑥(
∵@ D M𝐴𝐵?@(@ D 𝐷𝐸⊥𝐴𝐵(
∴∠𝐴𝐷𝐸= 90°(𝐶𝐷= 𝐴𝐷= 𝐵𝐷(
∴∠𝐴= ∠𝐴𝐶𝐷(
∴sin∠𝐴= sin∠𝐴𝐶𝐷(
∴
𝐷𝐸
𝐴𝐸=
𝐸𝐻
𝐶𝐸(
∴
25𝑥
8−𝐶𝐸=
7𝑥
𝐶𝐸(
s
𝐶𝐸=
7
4(
∴𝐴𝐸= 8−
7
4 =
25
4 (
ÈYrs
cos∠𝐴=
𝐴𝐷
𝐴𝐸=
𝐴𝐶
𝐴𝐵(q𝐴𝐵= 2𝐴𝐷(
∴2𝐴𝐷2 =
25
4 × 8 = 50(
st𝐴𝐷= 5(
∴𝐷𝐸=
𝐴𝐸2−𝐴𝐷2 =
15
4 t
! B
@ÉNQHgVWXYH(À]\]^_H(§]\]>ÖG1H§HST(`Ê\
]>HST(ÌÍHÎÏgNHh
7 #3 *-#24-25 ./01·×Ø·Ù?-:;(y>𝐴𝐵𝐶𝐷?(M gG𝐶𝐷H?@(N gG𝐵𝐶H?@(®
Ò𝐴𝑀(𝐷𝑁f@ E(®Ò𝐶𝐸Ú(𝐴𝐷@ F J𝐸𝑀= 1(K𝐷𝐹HÔM#-
A
5
2
B
2 5
5
C
2 5
3
D
3 5
5
C
* ±y>ST[?@ÛÜ( △𝐴𝐷𝑀≌△𝐷𝐶𝑁(s∠𝐷𝐸𝑀= 90° kl\]^_Ý[
a𝐸𝑀= 1(𝐷𝐸U𝐷𝑀U𝐷𝐶U𝐷𝑁U𝑁𝐸HÔ o𝐷𝐹∥𝑁𝐶s² △𝐷𝐸𝐹∽△𝑁𝐸𝐶(±fÞ\]>ÆG
ßÂà𝐷𝐹HÔ
}∵~G>𝐴𝐵𝐶𝐷gy>(𝑀U𝑁*áMG𝐶𝐷U𝐵𝐶H?@(
∴𝐷𝐴= 𝐶𝐷= 𝐵𝐶(∠𝐴𝐷𝑀= ∠𝐷𝐶𝑁= 90°(
∴𝐷𝑀=
1
2𝐶𝐷=
1
2𝐵𝐶=
1
2𝐷𝐴= 𝐶𝑁(
< △𝐴𝐷𝑀¤ △𝐷𝐶𝑁?(
𝐴𝐷= 𝐷𝐶
∠𝐴𝐷𝑀= ∠𝐷𝐶𝑁
𝐷𝑀= 𝐶𝑁
∴△𝐴𝐷𝑀≌△𝐷𝐶𝑁(SAS)(
∴∠𝐷𝐴𝑀= ∠𝐶𝐷𝑁(
∴∠𝐷𝐸𝑀= ∠𝐷𝐴𝑀+ ∠𝐴𝐷𝑁= ∠𝐶𝐷𝑁+ ∠𝐴𝐷𝑁= ∠𝐴𝐷𝐶= 90°(
∵
𝐷𝐸
𝐸𝑀=
𝐴𝐷
𝐷𝑀= tan∠𝐴𝑀𝐷= 2(
∵𝐸𝑀= 1(
∴𝐷𝐸= 2𝐸𝑀= 2(
∴𝐷𝑀= 𝐶𝑁=
𝐸𝑀2 + 𝐷𝐸2 =
12 + 22 =
5(
∴𝐷𝐶= 2𝐷𝑀= 2 5(
∴𝐷𝑁=
𝐶𝑁2 + 𝐷𝐶2 =
( 5)
2 + (2 5)
2
= 5(
∴𝑁𝐸= 𝐷𝑁−𝐷𝐸= 5−2 = 3(
∵ 𝐷𝐹∥𝑁𝐶(
∴△𝐷𝐸𝐹∽△𝑁𝐸𝐶(
∴
𝐷𝐹
𝐶𝑁=
𝐷𝐸
𝑁𝐸=
2
3(
∴𝐷𝐹=
2
3𝐶𝑁=
2
3 ×
5 =
2 5
3
!
C
@ÉNQy>HSTUË`\]>HXSTUfÞ\]>HXSTZ[\]^_H
(hg±y>ST\]>Ë`fÞ(¯ÌfST¤7ÔÝâÇãä
8 #3 *-#2025·åæçè·-:;(éYêëìí îï<ðñïò(ïòó?!°Rôõö
÷øù(ú C Is EUF ô@Hû] αUβ *áM60°¤30°(Jüïý𝐶𝐷HÔM6cm(þ÷@ F fÆ
@ E ÿ¼R# -
A 3cm
B 3 3cm
C (3 3−3)cm
D (3 3 + 3)cm
C
* NQ§]\]>H(@𝐸𝐸𝑀⊥𝐶𝐷,𝐹𝑁⊥𝐶𝐷(klĀ(s²𝐶𝐹= 𝐶𝐷= 𝐶𝐸= 6
cm(±\]^_s²𝐶𝑀= 3 3(𝐶𝑁= 3(±7H¤āÝpvr
}
@𝐸𝐸𝑀⊥𝐶𝐷,𝐹𝑁⊥𝐶𝐷(oĀ(ra
𝐶𝐹= 𝐶𝐷= 𝐶𝐸= 6cm(𝐴𝐵⊥𝐶𝐷(
∴∠𝐸𝐶𝑀= 90°−𝛼= 30°,∠𝐹𝐶𝑁= 90°−𝛽= 60°(
∴𝐶𝑀= 𝐶𝐸⋅cos30° = 3 3(𝐶𝑁= 𝐶𝐹⋅cos60° = 3(
∴𝑀𝑁= 𝐶𝑀−𝐶𝑁= (3 3−3)cm(
! C
9 #3 *-#2025·Ă㥷-:;(rCDąĆć$í\Ĉ(ąĆćĉĊċ(®ÒČčĎďH
Đ]M30°(®ÒČčđò90°xćĒēËĉĊ(, k®ÒČč15cm#rĔ Û7-(ĕ\ĈąĆ
ćēËĉĊx(Ėė¼ÔÂĉĊċĘę# -cm
A 15 3
B 30 3
C 15 3−15
D
15 3
2 −
15
2
C
* NQR§]\]>H(c8de§]\]>H¦gh ĚĀ;#ě
-(±§]\]>*á𝐴𝐶,𝐷𝐹H(oþvrs
}
:; 1(®ÒČč𝐴𝐵= 15cm(®ÒČčĎďHĐ]∠𝐵= 30°(𝐴𝐶⊥𝐵𝐶(
∴, ĈĉĊċH¼ÔM𝐴𝐶= 𝐴𝐵⋅sin𝐵= 15 ×
1
2 = 7.5(cm)(
:; 2(®ÒČč𝐸𝐹= 15cm(®ÒČčĎďHĐ]#À]-∠𝐷𝐸𝐹= 180°−30°−90° = 60°(𝐷𝐹⊥𝐷𝐸(
∴, ĈĉĊxH¼ÔM𝐷𝐹= 𝐸𝐹⋅sin∠𝐷𝐸𝐹= 15 ×
3
2 =
15 3
2 (cm)(
∴ĕ\ĈąĆćēËĉĊx(Ėė¼ÔÂĉĊċĘęR2(𝐷𝐹−𝐴𝐶)
= 2 ×
15 3
2
−7.5
= 15 3−15(cm)(
!
C
10 #3 *-#24-25 Ĝ/0ĝ·Ğ3ğ5·Ù?-:;(<½§]ĠġÝ?(
a@𝐴1, 3 (Zè@ O M
Ģģ(Z𝑂𝐴MĤĥĚĦ( x ħĨĤħ@ B(®Ò𝐴𝐵 *áZ@ AUB MĢģ(Z𝐴𝐵MĤĥĚĦ(
ôĦ< ĩĪ@ C(®Ò𝑂𝐶 īA7𝑂𝐶Ĭè@ĭ÷Įįđ(,İįđ90°(K 2025 İįđ¯ı
÷(@ C HĠġM# -
A 2,2 3
B 2 3,2
C −2,−2 3
D −2 3,−2
C
* NOPQR;>ĠġUįđSTU§]\]>HSTU`G\]>HXST`ab
@(ymdefSTIJijgHh
:;
@ A 𝐴𝐻⊥𝑥ħ H(kl\]^_HX{s²∠𝐴𝑂𝐻= 60°(o;órs(§𝑂𝐶M7𝐴𝐵
HĴ§*(𝑂𝐵= 𝑂𝐴= 2(klVWXYs²𝐵𝐶= 𝐴𝐵=
(−2−1)2 + ( 3)
2
= 2 3(nĵ
∠𝑂𝐵𝐶= 90°(v İįđs² C @HÆ@M −2,2 3 tĶİs 2 İU 3 İU 4 İķī, 4 õ
õĸĹ(lþvr
}
:;
@ A 𝐴𝐻⊥𝑥ħ H(
∵𝐴1, 3 (
∴𝑂𝐴=
12 + ( 3)
2
= 2,tan∠𝐴𝑂𝐻=
𝐴𝐻
𝑂𝐻=
3(
∴∠𝐴𝑂𝐻= 60°(
o;órs(§𝑂𝐶M7𝐴𝐵HĴ§*(𝐴𝐶= 𝐵𝐶= 𝐴𝐵
∴𝑂𝐵= 𝑂𝐴= 2(
∴𝐴𝐶= 𝐵𝐶= 𝐴𝐵(𝐵−2(0 (∠𝐴𝐵𝑂= ∠𝐵𝐴𝑂(
∴△𝐴𝐵𝐶g`G\]>(
∴∠𝐴𝐵𝐶= 60°(
∴∠𝑂𝐶𝐵= 30°(
∵𝐵𝐶= 𝐴𝐵=
(−2−1)2 + ( 3)
2
= 2 3(
∵∠𝐴𝐵𝑂+ ∠𝐵𝐴𝑂= ∠𝐴𝑂𝐻= 60°(
∴∠𝐴𝐵𝑂= 30°(
∴∠𝑂𝐵𝐶= ∠𝐴𝐵𝐶+ ∠𝐴𝐵𝑂= 90°(
∴𝐶−2,2 3 (
∴ 1 İįđs² C @HÆ@M −2 3,−2 (
Ķİs
2 İįđs² C @HÆ@M 2,−2 3 (
3 İįđs² C @HÆ@M 2 3,2 (
4 İįđs² C @HÆ@M −2,2 3 (
5 İįđs² C @HÆ@M −2 3,−2 (
∴ķī, 4 õ õĸĹ(
∵2025 ÷ 4 = 506⋯⋯1(
∴ 2025 İįđ¯ı÷(@ C HĠġM −2 3,−2 (
!
C
ĺĻ#$ 6 '()* 18 *(,' 3 *-
11 #3 *-#2025·ľĿŀ5·-:;(<Ł>𝐴𝐵𝐶𝐷?(@𝐸,𝐹*ág𝐵𝐶,𝐶𝐷H?@(®Ò𝐴𝐸,𝐴𝐹 Ó
cos∠𝐸𝐴𝐹=
3
4(𝐴𝐸= 4(K𝐴𝐵HM
8 2
3
* þOPQRŁ>HST(§]\]>(c8deŁ>HST(Ë`\]>HX¤ST(
¨©ªÀ]\]^_HX{[VWXYp³´g¬Hh
𝐴𝐹,𝐵𝐶f@𝑀(@𝐸𝐸𝑁⊥𝐴𝐹@𝑁(u𝐵𝐸= 𝐶𝐸= 𝑎(K𝐴𝐵= 𝐵𝐶= 𝐶𝐷= 𝐴𝐷= 2𝑎( △𝐴𝐵𝐸
¤ △𝐴𝐷𝐹Ë`s𝐴𝐸= 𝐴𝐹= 4(n △𝐴𝐷𝐹¤ △𝑀𝐶𝐹Ë`s𝐴𝐹= 𝑀𝐹= 4,𝐴𝐷= 𝐶𝑀= 2𝑎(K𝐸𝑀= 3𝑎(
Rt △𝐴𝐸𝑁s𝐴𝑁= 3,𝐹𝑁= 1,𝐸𝑁=
7(pqs𝑀𝑁= 5(wx<Rt △𝑀𝐸𝑁?oVWXY𝑎=
4 2
3 (ł
qrs𝐴𝐵H
}
𝐴𝐹,𝐵𝐶f@𝑀(@𝐸𝐸𝑁⊥𝐴𝐹@𝑁(:;
∵@𝐸g𝐵𝐶H?@(
∴u𝐵𝐸= 𝐶𝐸= 𝑎(K𝐵𝐶= 2𝑎(
∵~G>𝐴𝐵𝐶𝐷gŁ>(
∴𝐴𝐵= 𝐵𝐶= 𝐶𝐷= 𝐴𝐷= 2𝑎,𝐴𝐷∥𝐵𝐶,∠𝐵= ∠𝐷(
∵@𝐹g𝐶𝐷H?@(
∴𝐷𝐹= 𝐶𝐹= 𝑎(
∴𝐵𝐸= 𝐷𝐹(
< △𝐴𝐵𝐸¤ △𝐴𝐷𝐹?(
𝐴𝐵= 𝐴𝐷
∠𝐵= ∠𝐷
𝐵𝐸= 𝐷𝐹
(
∴△𝐴𝐵𝐸≌△𝐴𝐷𝐹(SAS)(
∴𝐴𝐸= 𝐴𝐹= 4(
∵𝐴𝐷∥𝐵𝐶(
∴∠𝐷𝐴𝐹= ∠𝑀,∠𝐷= ∠𝑀𝐶𝐹(
< △𝐴𝐷𝐹¤ △𝑀𝐶𝐹?(
∠𝐷𝐴𝐹= ∠𝑀
∠𝐷= ∠𝑀𝐶𝐹
𝐷𝐹= 𝐶𝐹
(
∴△𝐴𝐷𝐹≌△𝑀𝐶𝐹(AAS)(
∴𝐴𝐹= 𝑀𝐹= 4,𝐴𝐷= 𝐶𝑀= 2𝑎(
∴𝐸𝑀= 𝐶𝐸+ 𝐶𝑀= 3𝑎(
<Rt △𝐴𝐸𝑁?(cos∠𝐸𝐴𝐹=
𝐴𝑁
𝐴𝐸=
3
4(
∴𝐴𝑁=
3
4𝐴𝐸=
3
4 × 4 = 3(
∴𝐹𝑁= 𝐴𝐹−𝐴𝑁= 4−3 = 1(
oVWXYs
𝐸𝑁=
𝐴𝐸2−𝐴𝑁2 =
42−32 =
7(
<Rt △𝑀𝐸𝑁?(𝑀𝑁= 𝑀𝐹+ 𝐹𝑁= 5(
oVWXYs
𝐸𝑀=
𝐸𝑁2 + 𝑀𝑁2 =
( 7)
2 + 52 = 4 2(
∴3𝑎= 4 2(
∴𝑎=
4 2
3 (
∴𝐴𝐵= 2𝑎=
8 2
3 (
M
8 2
3
12 #3 *-#24-25 Ĝ/0ĝ·¸¹º»·ÙŃ-:;(<Rt △𝐴𝐵𝐶?(∠𝐴𝐶𝐵= 90°(∠𝐵= 60°(𝐵𝐶= 2(
A △𝐴𝐵𝐶Ĭ@ C ĭ÷Įįđs² △𝐴′𝐵′𝐶 Ó𝐵′ńŅF<𝐴𝐵1(®Ò𝐴𝐴′(°𝐴𝐴′H?@ D(®Ò𝐵′𝐷(K
𝐵′𝐷HM
7
* oįđSTr △𝐵𝐵′𝐶`G\]>(𝐵′𝐴= 𝐵′𝐶= 𝐵𝐶= 2 n𝐵′𝐴′M𝐴𝐶H?Ĵ(K𝐴𝐴′ =
𝐴′𝐶= 𝐴𝐶= 2 3(wx △𝐴𝐵′𝐷M§]\]>(ņ𝐴𝐷=
1
2𝐴𝐴′ =
3(𝐴𝐵′ = 2(VWXYr𝐵′𝐷H
}
o∠𝐵=60°[įđSTra𝐵𝐶= 𝐵′𝐶(
∴△𝐵𝐵′𝐶M`G\]>
∴∠𝐵′𝐶𝐴= 90°−60° = 30° = ∠𝐵𝐴𝐶(
∴𝐵′𝐴= 𝐵′𝐶= 𝐵𝐶= 2
ņ∠𝐴𝐶𝐴′ = 90°−∠𝐵′𝐶𝐴= 60°(∠𝐶𝐴′𝐵′ = ∠𝐵𝐴𝐶= 30°(
∴𝐵′𝐴′ ⊥𝐴𝐶(
∴𝐵′𝐴′M𝐴𝐶H?Ĵ(
∴𝐴𝐴′ = 𝐴′𝐶= 𝐴𝐶= 2 × tan60° = 2 3
∴∠𝐴′𝐴𝐶= ∠𝐴′𝐶𝐴= 60°(
∴∠𝐵′𝐴𝐷= ∠𝐵𝐴𝐶+ ∠𝐴′𝐴𝐶= 90°(
ņ D M𝐴𝐴′?@(
∴𝐴𝐷=
1
2𝐴𝐴′ =
3(
∴𝐵′𝐷=
𝐴𝐵′2 + 𝐴𝐷2 =
22 + ( 3)
2
=
7
M
7
@ÉNQR;>įđHST(§]\]>HST(«Ň]H\]^_L(VWXY(`G\]>
HST(`Ê\]>HST(ňʼndeŊŋSTgh
13 #3 *-#2025·2÷ -'Ō ōŎŏŃjŐ;őŒĔő(wxŐœŔŕ :;(;őŒ(;?@
A I)<Ŗō(;?@ B I)HÃŗ¹20∘ď(ÓŘřŖō2km(œ(;?@ C I)<ŖōHÃŗ25°ď¤
;őŒHÃŗ62°ďHŚI(śŜ(ŖōœHŘřŝM km #¯Şşm²0.1km._l
sin37° ≈0.6,cos37° ≈0.8,tan37° ≈0.75, 2 ≈1.414-
3.3
* NOPQR§]\]>H(@ A 𝐴𝐷⊥𝐵𝐶@ D(K∠𝐴𝐷𝐵= ∠𝐴𝐷𝐶= 90°(
𝐶𝐷≈
4 2
3 (rs𝐵𝐶= 𝐵𝐷+ 𝐶𝐷≈
7 2
3 ≈3.3(km)
}:;(@ A 𝐴𝐷⊥𝐵𝐶@ D(K∠𝐴𝐷𝐵= ∠𝐴𝐷𝐶= 90°(
oĀ(s ∠𝐵𝐴𝐶= 180°−20°−62° = 98°(∠𝐴𝐵𝐷= 20° + 25° = 45°(
∴∠𝐶= 180°−∠𝐵𝐴𝐶−∠𝐴𝐵𝐷= 37°.
∵𝐴𝐵= 2(
∴𝐵𝐷= 𝐴𝐵⋅cos∠𝐴𝐵𝐷=
2(𝐴𝐷= 𝐵𝐷=
2
∵tan𝐶≈0.75(
∴𝐶𝐷= 𝐴𝐷
tan𝐶≈4 2
3 (
∴𝐵𝐶= 𝐵𝐷+ 𝐶𝐷≈
7 2
3 ≈3.3(km)
vŖōœHŘřŝM3.3km
M
3.3
14 #3 *-#2025·ĂŠ5·\-:;(<yšG>𝐴𝐵𝐶𝐷𝐸?(®Ò𝐶𝐸(@ F M𝐶𝐸H?@(®Ò𝐷𝐹Ú
(G𝐴𝐵@ G(K
𝐷𝐹
𝐹𝐺HLM
#¯Şşm²0.1 _l
sin36° ≈0.59,cos36° ≈0.81,tan
36° ≈0.73,sin72° ≈0.95,tan72° ≈3.08-
0.6
* ŢOPQyţG>HST(\]>(=>HX(YĀ(ÎÏ(ŤÌªŊŋ
ab@gh
@ A 𝐴𝐻⊥𝐸𝐶(klţG>IJ]¤XY[ťSTs∠𝐷𝐸𝐶= 36°,∠𝐴𝐸𝐻= 72°(no=>HX¤ST
s~G>𝐻𝐹𝐴𝐺M=>(𝐴𝐻= 𝐹𝐺(u𝐸𝐷= 𝐴𝐸= 𝑥(±y^_s𝐷𝐹= 0.59𝑥(𝐴𝐻= 𝐹𝐺= 0.95𝑥(
vr
}
@ A 𝐴𝐻⊥𝐸𝐶(:;
∵yšG>𝐴𝐵𝐶𝐷𝐸(
∴∠𝐸𝐷𝐶= ∠𝐴𝐸𝐷= ∠𝐸𝐴𝐵=
180°×(5−2)
5
= 108°,𝐷𝐸= 𝐷𝐶= 𝐴𝐸(
∴∠𝐷𝐸𝐶=
180°−108°
2
= 36°,∠𝐴𝐸𝐻= 108°−36° = 72°(
∴∠𝐸𝐴𝐵+ ∠𝐴𝐸𝐻= 180°(
∴𝐸𝐶∥𝐴𝐵(
∵𝐷𝐸= 𝐷𝐶(@ F M𝐶𝐸H?@(
∴𝐷𝐹⊥𝐸𝐶,𝐷𝐺⊥𝐴𝐵(
∴∠𝐴𝐻𝐹= ∠𝐻𝐹𝐺= ∠𝐹𝐺𝐴= 90°(
∴~G>𝐻𝐹𝐴𝐺M=>(
∴𝐴𝐻= 𝐹𝐺(
u𝐸𝐷= 𝐴𝐸= 𝑥(
∴sin∠𝐷𝐸𝐹= sin36° =
𝐷𝐹
𝐷𝐸(
∴𝐷𝐹= 0.59𝑥(
∴sin∠𝐴𝐸𝐻= sin72° =
𝐴𝐻
𝐴𝐸(
∴𝐴𝐻= 𝐹𝐺= 0.95𝑥(
∴
𝐷𝐹
𝐹𝐺=
0.59𝑥
0.95𝑥≈0.6(
M
0.6
15 #3 *-#2025·¹Ŧŧ·\-:;(<~G>𝐴𝐵𝐶𝐷?(𝐴𝐵= 𝐵𝐶(𝐴𝐶⊥𝐶𝐷(𝐵𝐷*∠𝐴𝐷𝐶(𝐴𝐶
𝐵𝐷f@𝐸(J𝐶𝐷= 3(𝐴𝐶= 4(K7𝐵𝐸HM
5
2
* N×@Q`Ê\]>HSTU]*HSTUË`\]>HXSTUVWXYU§
]\]>`ab(ymŨũęÎÏgHh
𝐸𝐿⊥𝐷𝐴@𝐿(q𝐴𝐶⊥𝐶𝐷(𝐶𝐷= 3(𝐴𝐶= 4(s𝐷𝐴= 5(o𝐵𝐷*∠𝐴𝐷𝐶(s𝐸𝐶= 𝐸𝐿(osin∠𝐶𝐴𝐷=
𝐸𝐿
𝐴𝐸=
𝐶𝐷
𝐷𝐴=
3
5(s𝐸𝐶= 𝐸𝐿=
3
5𝐴𝐸(K𝐴𝐸+
3
5𝐴𝐸= 4(s𝐴𝐸=
5
2(K𝐸𝐶 =
3
2(𝐵𝐻⊥𝐷𝐴@𝐻,𝐵𝐹⊥𝐷𝐶
𝐷𝐶H@𝐹(K𝐵𝐻= 𝐵𝐹(rRt △𝐴𝐵𝐻≌Rt △𝐶𝐵𝐹(s∠𝐵𝐴𝐻= ∠𝐵𝐶𝐹(ãä
∠𝐴𝐷𝐶+ ∠𝐴𝐵𝐶= 2∠𝐶𝐷𝐸+ ∠𝐴𝐵𝐶= 180°(o𝐴𝐵= 𝐵𝐶(s∠𝐵𝐴𝐸= ∠𝐵𝐶𝐸(K2∠𝐵𝐴𝐸+ ∠𝐴𝐵𝐶= 180°(r
∠𝐵𝐴𝐸= ∠𝐶𝐷𝐸(K∠𝐴𝐵𝐸= ∠𝐸𝐶𝐷= 90°(o
𝐵𝐸
𝐴𝐵= tan∠𝐵𝐴𝐸= tan∠𝐶𝐷𝐸=
𝐸𝐶
𝐶𝐷=
1
2(s𝐴𝐵 = 2𝐵𝐸(
Z𝐴𝐸=
5𝐵𝐸=
5
2(s𝐵𝐸=
5
2 (gs²H
}𝐸𝐿⊥𝐷𝐴@𝐿(K∠𝐴𝐿𝐸= 90°(
∵𝐴𝐶⊥𝐶𝐷(𝐶𝐷= 3(𝐴𝐶= 4(
∴∠𝐴𝐶𝐷= 90°(
∴𝐷𝐴=
𝐶𝐷2 + 𝐴𝐶2 =
32 + 42 = 5(
∵𝐵𝐷*∠𝐴𝐷𝐶(𝐴𝐶𝐵𝐷f𝐵@𝐸(Ó𝐸𝐶⊥𝐷𝐶(𝐸𝐿⊥𝐷𝐴(
∴𝐸𝐶= 𝐸𝐿,∠𝐴𝐷𝐶= 2∠𝐶𝐷𝐸(
∵sin∠𝐶𝐴𝐷=
𝐸𝐿
𝐴𝐸=
𝐶𝐷
𝐷𝐴=
3
5(
∴𝐸𝐶= 𝐸𝐿=
3
5𝐴𝐸(
∴𝐴𝐸+
3
5𝐴𝐸= 4(
s𝐴𝐸=
5
2(
∴𝐸𝐶=
3
5 ×
5
2 =
3
2(
𝐵𝐻⊥𝐷𝐴@𝐻(𝐵𝐹⊥𝐷𝐶𝐷𝐶H@𝐹(K∠𝐴𝐻𝐵= ∠𝐹= 90°(𝐵𝐻= 𝐵𝐹(
<Rt △𝐴𝐵𝐻¤ Rt △𝐶𝐵𝐹?(
𝐴𝐵= 𝐶𝐵
𝐵𝐻= 𝐵𝐹
,
∴Rt △𝐴𝐵𝐻≌Rt △𝐶𝐵𝐹(HL)(
∴∠𝐵𝐴𝐻= ∠𝐵𝐶𝐹(
∵∠𝐵CD +∠𝐵𝐶𝐹= 180°(
∴∠𝐵𝐶𝐷+ ∠𝐵𝐴𝐻= 180°(
∴∠𝐴𝐷𝐶+ ∠𝐴𝐵𝐶= 360°−(∠𝐵𝐶𝐷+ ∠𝐵𝐴𝐻) = 180°(
∴2∠𝐶𝐷𝐸+ ∠𝐴𝐵𝐶= 180°(
∵𝐴𝐵= 𝐵𝐶(
∴∠𝐵𝐴𝐸= ∠𝐵𝐶𝐸(
∴2∠𝐶𝐷𝐸+ ∠𝐴𝐵𝐶= 180°(
∴2∠𝐵𝐴𝐸+ ∠𝐴𝐵𝐶= 2∠𝐶𝐷𝐸+ ∠𝐴𝐵𝐶(
∴∠𝐵𝐴𝐸= ∠𝐶𝐷𝐸(
∴∠𝐴𝐵𝐸= ∠𝐴𝐸𝐷−∠𝐵𝐴𝐸= ∠𝐴𝐸𝐷−∠𝐶𝐷𝐸= ∠𝐸𝐶𝐷= 90°(
∴
𝐵𝐸
𝐴𝐵= tan∠𝐵𝐴𝐸= tan∠𝐶𝐷𝐸=
𝐸𝐶
𝐶𝐷=
3
2
3 =
1
2(
∴𝐴𝐵= 2𝐵𝐸(
∵𝐴𝐸=
𝐴𝐵2 + 𝐵𝐸2 =
(2𝐵𝐸)2 + 𝐵𝐸2 =
5𝐵𝐸=
5
2(
∴𝐵𝐸=
5
2
M
5
2
16 #3 *-#2025·ĞÃŪū·\-:;(<Rt △𝐴𝐵𝐶?(∠𝐵𝐴𝐶= 90°(∠𝐴𝐶𝐵= 30°(𝐴𝐵= 𝑎(@ D <
G𝐴𝐶1ªò#Ŭ@ A(C ×Ì-(Z𝐵𝐷MGy>𝐵𝐷𝐸𝐹(E@ A <y>𝐵𝐷𝐸𝐹IJ(®Ò𝐸𝐶(K
#1-ĕ𝐶𝐷= 2𝐴𝐷÷(∠𝐴𝐷𝐸=
t
#2-@𝐹²§𝐴𝐵HŘřM
t
#3- △𝐶𝐷𝐸½¾HŭŮLg
30°
𝑎
3
8𝑎2
* þQRy>HST(Ë`\]>HX[ST(À]\]^_` #1-±§]\]>H
STs𝐴𝐶=
3𝑎(𝐴𝐷=
3
3 𝑎(kltan∠𝐵𝐷𝐴=
𝐴𝐵
𝐴𝐷s∠𝐴𝐷𝐵= 60°(nkly>HSTs
∠𝐴𝐷𝐸= 30°t#2-@ F 𝐹𝐺⊥𝐴𝐵 G( △𝐹𝐺𝐵≌△𝐵𝐴𝐷(vrs𝐹𝐺= 𝐴𝐵= 𝑎t#3-@ E
𝐸𝐻⊥𝐴𝐶 H( △𝐷𝐻𝐸≌△𝐵𝐴𝐷(s²𝐸𝐻= 𝐴𝐷(±\]^_𝐴𝐷U𝐴𝐶(kl𝑆△𝐶𝐷𝐸=
1
2
𝐶𝐷⋅𝐻𝐸(±ůĨ_HSTvr
}
#1-∵∠𝐵𝐴𝐶= 90°(∠𝐴𝐶𝐵= 30°(𝐴𝐵= 𝑎(
∴𝐵𝐶= 2𝐴𝐵= 2𝑎(
∴𝐴𝐶=
𝐵𝐶2−𝐴𝐵2 =
3𝑎(
∵𝐶𝐷= 2𝐴𝐷(
∴𝐴𝐷=
1
3𝐴𝐶=
3
3 𝑎(
∵tan∠𝐵𝐷𝐴=
𝐴𝐵
𝐴𝐷=
𝑎
3
3 𝑎=
3(
∴∠𝐴𝐷𝐵= 60°(
∵y>𝐵𝐷𝐸𝐹(
∴∠𝐴𝐷𝐸= 90°−60° = 30°t
M
30°t
#2-@ F 𝐹𝐺⊥𝐴𝐵 G(K∠𝐹𝐺𝐵= 90°(
∵~G>𝐵𝐷𝐸𝐹gy>(
∴𝐵𝐹= 𝐵𝐷(∠𝐷𝐵𝐹= ∠𝐵𝐴𝐶= 90°(
∴∠𝐺𝐹𝐵+ ∠𝐹𝐵𝐺= ∠𝐹𝐵𝐺+ ∠𝐴𝐵𝐷= 90°(
∴∠𝐺𝐹𝐵= ∠𝐴𝐵𝐷(
∴△𝐹𝐺𝐵≌△𝐵𝐴𝐷(
∴𝐹𝐺= 𝐴𝐵= 𝑎(v@𝐹²§𝐴𝐵HŘřM𝑎t
M
𝑎t
#3-@ E 𝐸𝐻⊥𝐴𝐶 H(K∠𝐻= 90°(
∵~G>𝐵𝐷𝐸𝐹gy>(
∴𝐷𝐸= 𝐵𝐷(∠𝐵𝐷𝐸= ∠𝐵𝐴𝐶= 90°(
∴∠𝐴𝐷𝐸+ ∠𝐴𝐷𝐵= ∠𝐴𝐷𝐵+ ∠𝐴𝐵𝐷= 90°(
∴∠𝐴𝐷𝐸= ∠𝐴𝐵𝐷(
∴△𝐷𝐻𝐸≌△𝐵𝐴𝐷(
∴𝐸𝐻= 𝐴𝐷(
∵𝐴𝐷= 𝐴𝐵⋅tan∠𝐴𝐵𝐷= 𝑎⋅tan∠𝐴𝐵𝐷(𝐴𝐶=
3𝑎(
∴𝐶𝐷= 𝐴𝐶−𝐴𝐷= ( 3−tan∠𝐴𝐵𝐷)𝑎(
∴𝑆△𝐶𝐷𝐸=
1
2𝐶𝐷⋅𝐻𝐸
= 1
2 ( 3−tan∠𝐴𝐵𝐷)𝑎⋅𝑎⋅tan∠𝐴𝐵𝐷
= −
1
2 tan∠𝐴𝐵𝐷−
3
2
2
+
3
8 𝑎2(
∴ĕtan∠𝐴𝐵𝐷−
3
2 = 0÷( △𝐶𝐷𝐸½¾HŭŮLg
3
8𝑎2
M
3
8𝑎2
Ⅱ
\ #$ 8 '()* 72 *-
17 #6 *-#24-25 Ĝ/0ĝ·ÃŲ·ÙŃ-:;(<Rt △𝐴𝐵𝐶?(∠𝐶𝐴𝐵= 90°(sin𝐶=
3
5(𝐴𝐶= 8(𝐵𝐷
*∠𝐶𝐵𝐴𝐴𝐶G@𝐷
(1)§Òų7𝐴𝐵H
t
(2)𝐷@𝐷𝐸⊥𝐵𝐶@𝐸(ŴË;>(Ú7𝐴𝐷H
(1)6
(2);>ě t3
* N×@QRÀ]\]^_(VWXY(]*HST(c8ªfab(_>¯Ì(g
Hh
#1-±sin𝐶=
3
5s
𝐴𝐵
𝐵𝐶=
3
5(u𝐴𝐵= 3𝑥(𝑥> 0)(𝐵𝐶= 5𝑥(n±VWXYŵó(ó𝑥HL(r
s7𝐴𝐵Ht
#2-±]*HST(ã𝐴𝐷= 𝐸𝐷(n±𝑆△𝐴𝐵𝐶= 𝑆△𝐷𝐴𝐵+ 𝑆△𝐷𝐵𝐶(ŵót
}#1-<Rt △𝐴𝐵𝐶?(∠𝐶𝐴𝐵= 90°(𝐴𝐶= 8(
∴sin𝐶=
𝐴𝐵
𝐵𝐶=
3
5(
u𝐴𝐵= 3𝑥(𝑥> 0)(𝐵𝐶= 5𝑥(
oVWXYs𝐴𝐵2 +𝐴𝐶2 = 𝐵𝐶2(
∴(3𝑥)2 + 82 = (5𝑥)2(
s𝑥= 2(
∴𝐴𝐵= 3𝑥= 6(𝐵𝐶= 5𝑥= 10(
∴7𝐴𝐵HM 6
M
6
#2-:;
∵∠𝐶𝐴𝐵= 90°(
∴𝐷𝐴⊥𝐴𝐵(
∵𝐷𝐸⊥𝐵𝐶,𝐷𝐴⊥𝐴𝐵(𝐵𝐷*∠𝐶𝐵𝐴(
∴𝐴𝐷= 𝐸𝐷(
∵𝐴𝐵= 6(𝐴𝐶= 8(𝐵𝐶= 10(
∴𝑆△𝐴𝐵𝐶=
1
2 ⋅𝐴𝐵⋅𝐴𝐶=
1
2 × 6 × 8 = 24(𝑆△𝐷𝐴𝐵=
1
2𝐴𝐵⋅𝐴𝐷= 3𝐴𝐷(𝑆△𝐷𝐵𝐶=
1
2𝐵𝐶⋅𝐷𝐸= 5𝐷𝐸= 5𝐴𝐷(
ņ ∵𝑆△𝐴𝐵𝐶= 𝑆△𝐷𝐴𝐵+ 𝑆△𝐷𝐵𝐶(
∴24 = 3𝐴𝐷+ 5𝐴𝐷(
∴𝐴𝐷= 3(
∴7𝐴𝐷HM 3
18 #6 *-#2025·ÃŲ·?Ä-:;(< △𝐴𝐵𝐶?(D(E *áM𝐴𝐵(𝐴𝐶H?@(𝐷𝐹⊥𝐵𝐶(ĴŶM
F(@ G <𝐷𝐸H1(𝐷𝐺= 𝐹𝐶
(1)
~G>𝐷𝐹𝐶𝐺g=>t
(2)J∠𝐵= 45°(𝐷𝐹= 3(𝐷𝐺= 5(𝐵𝐶¤𝐴𝐶H
(1)ě
(2)𝐵𝐶= 8(𝐴𝐶= 2 10
* NOPQR=>HX(\]>?ŷXY(VWXY(§]\]>(cafabg
Hh
#1-o\]>?ŷXYrs𝐷𝐸∥𝐶𝐹(v𝐷𝐺∥𝐶𝐹(Kr~G>𝐷𝐹𝐶𝐺g~G>(no𝐷𝐹⊥𝐵𝐶(
vr~G>𝐷𝐹𝐶𝐺g=>t
#2-𝐶𝐹= 5(Rt △𝐵𝐷𝐹s²𝐵𝐷= 3 2(𝐵𝐹= 3(K𝐵𝐶= 𝐵𝐹+ 𝐶𝐹= 8to7?@HX{rs
𝐴𝐵= 2𝐵𝐷= 6 2t@ A 𝐴𝐻⊥𝐵𝐶 H(Rt △𝐴𝐵𝐻s²𝐴𝐻= 6(𝐵𝐻= 6(K𝐶𝐻= 𝐵𝐶−𝐵𝐻= 2(n
±VWXvr𝐴𝐶H
}#1-
∵D(E *áM𝐴𝐵(𝐴𝐶H?@(
∴𝐷𝐸g △𝐴𝐵𝐶H?ŷ(
∴𝐷𝐸∥𝐶𝐹(v𝐷𝐺∥𝐶𝐹(
∵𝐷𝐺= 𝐹𝐶(
∴~G>𝐷𝐹𝐶𝐺g~G>(
ņ∵𝐷𝐹⊥𝐵𝐶(
∴~G>𝐷𝐹𝐶𝐺g=>t
#2-
∵𝐷𝐺= 5(
∴𝐶𝐹= 𝐷𝐺= 5t
∵𝐷𝐹⊥𝐵𝐶(
∴∠𝐷𝐹𝐵= 90°(
<Rt △𝐵𝐷𝐹?(∠𝐵= 45°(𝐷𝐹= 3(
∴𝐵𝐷=
𝐷𝐹
sin𝐵=
3
sin45° = 3 2(𝐵𝐹=
𝐷𝐹
tan𝐵=
3
tan45° = 3(
∴𝐵𝐶= 𝐵𝐹+ 𝐶𝐹= 8t
∵@ D M𝐴𝐵H?@(
∴𝐴𝐵= 2𝐵𝐷= 6 2t
:;(@ A 𝐴𝐻⊥𝐵𝐶 H(
<Rt △𝐴𝐵𝐻?(𝐴𝐻= 𝐴𝐵⋅sin𝐵= 6 2 ⋅sin45° = 6(𝐵𝐻= 𝐴𝐵⋅cos𝐵= 6 2 ⋅cos45° = 6(
∴𝐶𝐻= 𝐵𝐶−𝐵𝐻= 2(
<Rt △𝐴𝐻𝐶?(oVWXYs𝐴𝐶=
𝐴𝐻2 + 𝐶𝐻2 =
62 + 22 = 2 10
19 #8 *-#24-25 ./0ĝ·Ÿ5Ź{·Ù?-:;(𝐴𝐶M=>𝐴𝐵𝐶𝐷HÆ](𝐴𝐶H?@𝑂𝐴𝐶HĴ(
*á𝐵𝐶(𝐴𝐷𝐸(𝐹(®Ò𝐴𝐸(𝐶𝐹
(1)
~G>𝐴𝐸𝐶𝐹MŁ>t
(2)J𝐴𝐷= 6(𝐴𝐵= 4(cos∠𝐶𝐹𝐷HL
(1)}ě
(2)
5
13
* #1- △𝑂𝐹𝐴≌△𝑂𝐸𝐶(AAS)(s²𝑂𝐹= 𝑂𝐸(wx¯Ì𝐴𝐶⊥𝐸𝐹vr~G>𝐴𝐸𝐶𝐹MŁ
>t
#2-u𝐷𝐹= 𝑥(o#1-a
𝐶𝐹= 𝐴𝐹= 6−𝑥(klVWXY𝐷𝐹=
5
3(wxklźHX{vr
}#1- ∵~G>𝐴𝐵𝐶𝐷M=>(
∴𝐴𝐷∥𝐵𝐶(
∴∠𝐹𝐴𝑂= ∠𝐸𝐶𝑂(∠𝑂𝐹𝐴= ∠𝑂𝐸𝐶(
∵𝑂M𝐴𝐶?@(
∴𝑂𝐴= 𝑂𝐶(
∴△𝑂𝐹𝐴≌△𝑂𝐸𝐶(AAS)(
∴𝑂𝐹= 𝑂𝐸(
∴~G>𝐴𝐸𝐶𝐹g~G>(
∵𝐴𝐶⊥𝐸𝐹(
∴~G>𝐴𝐸𝐶𝐹MŁ>t
#2- ∵~G>𝐴𝐵𝐶𝐷M=>(𝐴𝐷= 6(𝐴𝐵= 4(
∴𝐶𝐷= 𝐴𝐵= 4(
u𝐷𝐹= 𝑥
o#1-a
𝐶𝐹= 𝐴𝐹= 𝐴𝐷−𝐷𝐹= 6−𝑥(
Rt △𝐶𝐷𝐹?(𝐷𝐹2 +𝐶𝐷2 = 𝐶𝐹2(
v𝑥2 + 42 = (6−𝑥)2(
s𝑥=
5
3
∴𝐷𝐹=
5
3(𝐶𝐹= 6−
5
3 =
13
3 (
∴cos∠𝐶𝐹𝐷=
𝐷𝐹
𝐹𝐶=
5
3
13
3
=
5
13
@ÉþQR=>HST(VWXY(Ë`\]>HST¤X(Ł>HX`ab(Hh
gdeZ1ab@
20 #8 *-#2025·ĂŻ·-:;(y>𝐴𝐵𝐶𝐷HGM 3(EUF *ágG𝐴𝐷,𝐴𝐵1H@(®Ò
𝐶𝐸U𝐸𝐹U𝐶𝐹
(1):; 1(J𝐷𝐸= 1.8(ĕ∠𝐶𝐸𝐹= 90°÷(tan∠𝐸𝐶𝐹HLt
(2):; 2(J𝐴𝐸= 𝐴𝐹(𝐶𝐹𝐷𝐴H@ G(E M𝐷𝐺H?@(tan∠𝐸𝐶𝐹HLt
(3):; 3(J𝐷𝐸= 1(tan∠𝐸𝐶𝐹=
3
5(𝐵𝐹H
(1)
2
5
(2)
2
4
(3)𝐵𝐹=
18
7
* #1-ij𝐴𝐸= 𝐴𝐷−𝐷𝐸= 3−1.8 = 1.2( △𝐴𝐹𝐸∽△𝐷𝐸𝐶(s²
𝐸𝐹
𝐶𝐸=
𝐴𝐸
𝐶𝐷=
1.2
3 =
2
5(w
x±yzHX{vrt
#2-u𝐴𝐸= 𝐴𝐹= 𝑥(K𝐵𝐹= 𝐷𝐸= 3−𝑥(ż𝐴𝐺= 𝐺𝐸−𝐴𝐸= 3−2𝑥(wx △𝐺𝐴𝐹∽△𝐶𝐵𝐹(s
²
𝐴𝐺
𝐵𝐶=
𝐴𝐹
𝐵𝐹(¶Ž𝐴𝐸= 𝐴𝐹=
6−3 2
2
= 3−
3
2 2(𝐺𝐸= 𝐷𝐸= 3−𝑥=
3
2 2(𝐶𝐸=
𝐷𝐸2 + 𝐶𝐷2 =
3
2 6(:;
(@ G 𝐺𝐻⊥𝐶𝐸𝐶𝐸@ H( △𝐺𝐸𝐻∽△𝐶𝐸𝐷(s²
𝐺𝐻
𝐶𝐷=
𝐸𝐻
𝐷𝐸=
𝐺𝐸
𝐶𝐸(wx¶Ž
𝐺𝐻=
3(𝐸𝐻=
6
2 (𝐻𝐶= 𝐻𝐸+ 𝐶𝐸= 2 6(wxklyzHX{vrt
#3-:;(𝐷𝐴(𝐶𝐹@ G(@ G 𝐺𝐻⊥𝐶𝐸𝐶𝐸@ H(ij
𝐴𝐸= 𝐴𝐷−𝐷𝐸= 3−1 = 2(𝐸𝐶=
𝐷𝐸2 + 𝐶𝐷2 =
10(u𝐺𝐻= 3𝑥(𝐻𝐶= 5𝑥( △𝐺𝐸𝐻∽△𝐶𝐸𝐷(s
²
𝐺𝐻
𝐶𝐷=
𝐸𝐻
𝐷𝐸=
𝐺𝐸
𝐶𝐸(𝐺𝐸=
5
2(wx △𝐺𝐴𝐹∽△𝐶𝐵𝐹(s²
𝐴𝐺
𝐵𝐶=
𝐴𝐹
𝐵𝐹(¶_vr
}#1-
∵~G>𝐴𝐵𝐶𝐷gy>(
∴∠𝐴= ∠𝐷= 90° = ∠𝐶𝐸𝐹(𝐴𝐵= 𝐴𝐷= 𝐶𝐷= 3(
∴∠𝐴𝐹𝐸+ ∠𝐴𝐸𝐹= ∠𝐶𝐸𝐷+ ∠𝐴𝐸𝐹(𝐴𝐸= 𝐴𝐷−𝐷𝐸= 3−1.8 = 1.2(
∴∠𝐴𝐹𝐸= ∠𝐶𝐸𝐷(
∴△𝐴𝐹𝐸∽△𝐷𝐸𝐶(
∴
𝐸𝐹
𝐶𝐸=
𝐴𝐸
𝐶𝐷=
1.2
3 =
2
5(
∵∠𝐶𝐸𝐹= 90°(
∴tan∠𝐸𝐶𝐹=
𝐸𝐹
𝐸𝐶=
2
5t
#2-
∵𝐴𝐸= 𝐴𝐹(
∴u𝐴𝐸= 𝐴𝐹= 𝑥(
∴𝐵𝐹= 𝐷𝐸= 3−𝑥(
∵E M𝐷𝐺H?@(
∴𝐺𝐸= 𝐷𝐸= 3−𝑥(
∴𝐴𝐺= 𝐺𝐸−𝐴𝐸= 3−2𝑥(
∵𝐴𝐷∥𝐵𝐶(
∴△𝐺𝐴𝐹∽△𝐶𝐵𝐹(
∴
𝐴𝐺
𝐵𝐶=
𝐴𝐹
𝐵𝐹(v
3−2𝑥
3
=
𝑥
3−𝑥(
ĖYs(2𝑥2−12𝑥+ 9 = 0(
s
𝑥=
6−3 2
2
ž𝑥=
6+3 2
2
> 3#ſŐ-(
ƀƁë(𝑥=
6−3 2
2
gèóH(
∴𝐴𝐸= 𝐴𝐹=
6−3 2
2
= 3−
3
2 2(
∴𝐺𝐸= 𝐷𝐸= 3−𝑥=
3
2 2(
∴𝐶𝐸=
𝐷𝐸2 + 𝐶𝐷2 =
3
2 6(
:;(@ G 𝐺𝐻⊥𝐶𝐸𝐶𝐸@ H(
∴∠𝐻= ∠𝐷= 90°(
ņ∵∠𝐺𝐸𝐻= ∠𝐶𝐸𝐷(
∴△𝐺𝐸𝐻∽△𝐶𝐸𝐷(
∴
𝐺𝐻
𝐶𝐷=
𝐸𝐻
𝐷𝐸=
𝐺𝐸
𝐶𝐸(
∴
𝐺𝐻
3 =
𝐸𝐻
3
2 2 =
3
2 2
3
2 6(
∴𝐺𝐻=
3(𝐸𝐻=
6
2 (
∴𝐻𝐶= 𝐻𝐸+ 𝐶𝐸= 2 6(
∴tan∠𝐸𝐶𝐹=
𝐺𝐻
𝐻𝐶=
3
2 6 =
2
4 t
#3-
:;(𝐷𝐴(𝐶𝐹@ G(@ G 𝐺𝐻⊥𝐶𝐸𝐶𝐸@ H(
∵𝐷𝐸= 1(∠𝐷= 90°(𝐶𝐷= 3(
∴𝐴𝐸= 𝐴𝐷−𝐷𝐸= 3−1 = 2(𝐸𝐶=
𝐷𝐸2 + 𝐶𝐷2 =
10(
∵tan∠𝐸𝐶𝐹=
𝐺𝐻
𝐶𝐻=
3
5(
∴u𝐺𝐻= 3𝑥(𝐻𝐶= 5𝑥(
∴𝐻𝐸= 𝐻𝐶−𝐸𝐶= 5𝑥−10(
∵∠𝐻= ∠𝐷= 90°(∠𝐺𝐸𝐻= ∠𝐶𝐸𝐷(
∴△𝐺𝐸𝐻∽△𝐶𝐸𝐷(
∴
𝐺𝐻
𝐶𝐷=
𝐸𝐻
𝐷𝐸=
𝐺𝐸
𝐶𝐸(v
3𝑥
3 =
5𝑥−10
1
=
𝐺𝐸
10(
s
𝑥=
10
4 (𝐺𝐸=
5
2(
∴𝐺𝐴= 𝐺𝐸−𝐴𝐸=
5
2−2 =
1
2(
∵𝐴𝐷∥𝐵𝐶(
∴△𝐺𝐴𝐹∽△𝐶𝐵𝐹(
∴
𝐴𝐺
𝐵𝐶=
𝐴𝐹
𝐵𝐹(v
1
2
3 =
3−𝐵𝐹
𝐵𝐹(
∴𝐵𝐹=
18
7 (
ƀƁë(𝐵𝐹=
18
7 gèóH
@ÉþQRfÞ\]>HST¤X(y>HST(§]\]>(VWXY`ab(H
hgdeZ1ab@
21 #10 *-#2025 /ĞÃƂƃōƄµƅƆƇƈƉ?_ \ -ĞÃƂƊƋƌgƍƅƎƏHƐƑ
ƒƓ(¼ĻƔƕƖgƗƘƙHƚŢƒ :;1(𝐴𝐷U𝐵𝐸¢Ĵ§Ũ½Ó¼ÔfÈ ƐƑƛƜ<ŷč
@𝐶²𝐴𝐷<§HŘř𝐶𝐻= 3m(𝐵𝐶= 15m(þ÷∠𝐷𝐴𝐶= 36.87°t:;2(ĕƐƑƛƜƔƝƕƖ?@𝐹
÷(ńŅ∠𝐹𝐴𝐷= ∠𝐹𝐵𝐸= 60° ªòó?ýĒƞŬƟ#_l
sin36.87° ≈0.60(cos
36.87° ≈0.80(tan36.87° ≈0.75-
(1)𝐴𝐶Ht
(2)ƐƑƛƜú@𝐶Ɣ²@𝐹(ĝƠH¼Ôt
(3)<ú𝐹Ɣď𝐵Hó?(gơƢ<ƣõŷč𝑀(Es∠𝐴𝑀𝐵g§]Ƥ:ŞƢ<(ƥ@𝑀²@𝐵HŘřt
:ŞŬƢ<(ƥĵYo#¯ŞrƦƧk -
(1)𝐴𝐶HŝM5m
(2)ĝƠH¼Ô𝐻𝐼ŝM1m
(3)Ƣ<(@𝑀²@𝐵HŘřM(10−5 2)m
* Ng~G>HŤÌ(QRVWXY(§]\]>H(klŢH
aÛÜÚ¯Ì;
>ũęÍĕHÎÏgHh
#1-<Rt △𝐴𝐶𝐻?(±À]\]^_HX{p³´(vrt
#2-@𝐹𝐹𝐼⊥𝐴𝐷(ĴŶM𝐼(j±#1-H¯Õ¤7?@HX{rs𝐴𝐹= 5m(wx*á<Rt △𝐹𝐴𝐼
¤Rt △𝐴𝐶𝐻?(±À]\]^_HX{𝐴𝐼¤𝐴𝐻H(úq±7H¤āÝp³´(vrt
#3-:;2(®Ò𝐴𝐵(u𝐵𝑀= 𝑥m(K𝐴𝑀= (20−𝑥)m(klVWXYŵóvr
}#1-
<Rt △𝐴𝐶𝐻?(𝐶𝐻= 3m(∠𝐷𝐴𝐶= 36.87°(
∴sin36.87° =
𝐶𝐻
𝐴𝐶=
3
𝐴𝐶= 0.60(
∴𝐴𝐶= 5(m)(
∴𝐴𝐶HŝM5mt
#2-
:;1(@𝐹𝐹𝐼⊥𝐴𝐷(ĴŶM𝐼(
∵@𝐹MƕƖ?@(𝐴𝐶= 5m(𝐵𝐶= 15m(
∴𝐴𝐹=
1
2(𝐴𝐶+ 𝐵𝐶) =
1
2 × (5 + 15) = 10(𝑚)(
<Rt △𝐹𝐴𝐼?(∠𝐹𝐴𝐷= 60°(
∵cos60° =
𝐴𝐼
𝐴𝐹(
∴𝐴𝐼= 𝐴𝐹⋅cos60° = 10 ×
1
2 = 5(m)(
<Rt △𝐴𝐶𝐻?(oVWXYs
𝐴𝐻=
52−32 = 4(m)(
∴𝐻𝐼= 𝐴𝐼−𝐴𝐻= 5−4 = 1(m)(
∴ĝƠH¼Ô𝐻𝐼ŝM1mt
#3-
:;2(®Ò𝐴𝐵(
<Rt △𝐴𝐹𝐼?(𝐹𝐼=
102−52 = 5 3(m)(
∴𝐴𝐵= 2𝐹𝐼= 10 3(m)(
u𝐵𝑀= 𝑥m(K𝐴𝑀= (20−𝑥)m(
∵∠𝐴𝑀𝐵= 90°(
∴𝐴𝐵2 = 𝐴𝑀2 +𝐵𝑀2(
∴(10 3)
2
= (20−𝑥)2 + 𝑥2(
s
𝑥1 = 10 + 5 2 > 10(ſ)(𝑥2 = 10−5 2(
∴@𝑀²@𝐵HŘřM(10−5 2)m
22 #10 *-#2025·2ÃƨƩ· -:;go'y>ƪßH6 × 9 ¡(,õ'y>H£@ƫƬ¡@(
△𝐴𝐵𝐶H\õ£@ƭg¡@ Ʈ¥ƯÔH§ư<ƱX ¡?ēß~õĚ;ƲƳ(,õƲƳHĚŬsƴ
\Û
(1)<; 1 ?(jĚA7𝐵𝐴Ĭ@ A ĭ÷Įįđ90°xH7𝐷𝐴(n<𝐴𝐶1Ě@ E(Etan∠𝐴𝐵𝐸=
1
2t
(2)<; 2 ?(jĚA7𝐶𝐵Ĭ@ C Ƶ÷Įįđ2∠𝐴𝐶𝐵xH7𝐶𝐹(nĚ𝐹𝐻∥𝐴𝐵𝐴𝐶@ H
(1)ě
(2)ě
* #1-:;(°¡@ D(®Ò𝐴𝐷(°𝐴𝐷¡H@ P(®Ò𝐵𝑃𝐴𝐶 E(K7𝐷𝐴¤@ E
vMt
#2-:;(°¡@ TULUS(®Ò𝐵𝑆(®Ò𝑇𝐿Ú𝐵𝑆 F(®Ò𝐶𝐹(°¡@ MUN ®Ò𝑀𝑁𝐴𝐶
H(®Ò𝐹𝐻(K7𝐶𝐹(𝐹𝐻vM
}#1-
:;(°¡@ D(®Ò𝐴𝐷(°𝐴𝐷¡H@ P(®Ò𝐵𝑃𝐴𝐶 E(K7𝐷𝐴¤
@ E vMt
r𝐴𝐷= 𝐴𝐵(∠𝐵𝐴𝐷= 90°(K7𝐷𝐴vMt
r𝐴𝑃=
1
2𝐴𝐷=
1
2𝐴𝐵(Ktan∠𝐴𝐵𝑃=
𝐴𝑃
𝐴𝐵=
1
2(K@ E vMt
#2-
:;(°¡@ TULUS(®Ò𝐵𝑆(®Ò𝑇𝐿Ú𝐵𝑆 F(®Ò𝐶𝐹(°¡@ MUN ®Ò𝑀𝑁
𝐴𝐶 H(®Ò𝐹𝐻(K7𝐶𝐹(𝐹𝐻vMt
r𝐵𝐹⊥𝐹𝑇(𝐵𝐶= 𝐶𝑇(K𝐵𝐶= 𝐶𝐹(
r𝐴𝐶⊥𝐵𝐹(K∠𝐴𝐶𝐵= ∠𝐴𝐶𝐹(K7𝐶𝐹vMt
r𝐾𝑅∥𝐵𝐹∥𝑀𝑁Ó§𝐾𝑅²§𝐵𝐹HŘř`§𝑀𝑁²§𝐵𝐹H=>(
K𝐵𝐹*𝐴𝐶(ņí𝐴𝐶*𝐵𝐹(K~G>𝐴𝐵𝐻𝐹g~G>(K𝐹𝐻∥𝐴𝐵
@ÉNOPQRĚįđ;>(§]\]>(~G>HSTX(`Ê\]>HST
X(§]\]>HST``(c8de¡@;HƑƶ¤¦gHh
23 #12 *-#2025·1Ʒ· -:;gƣŨĝƸƹHƺ½;(ƣŤÌêƻ'ƪA¥ŎƼƽ<ƾƿǀ@
A I(ǁ¥ŎƼǂ²@𝐷I(𝐴𝐷ǃDŽ𝐵𝑅(s𝐴𝐷= 11.6Dž(þ÷<@𝐷Iņsƾƿ𝐴𝐵1H@𝐶H
û]M26.6° Òdžǁ¥ŎƼǂ²@𝐸I(𝐷𝐸⊥𝐴𝐷(𝐶𝐸ǃDŽ𝐵𝑅(s𝐷𝐸= 1.8Dž
(1)ƾƿ𝐴𝐵HƾÔt
(2)
aŨ½𝑄𝐴UŨĝƸƹH£DŽ𝐹𝐺ƭǃDŽ𝐵𝑅ÓLJLj²ǃDŽ𝐵𝑅HŘřf`(¥ŎƼú@𝐴ǂ²@𝑃I(
𝐴𝑃⊥𝐴𝐷(s𝐴𝑃= 16.4Dž(þ÷<@𝑃Is@𝐹Hû]M45°(<ŬljťNJNjnjHċǍĝ(í ǎ¼Ô
M 3 DžHǏƸǐơpŽüŨĝƸƹƤƥĵYo
#_l
sin26.6° ≈0.45(cos26.6° ≈0.89(tan26.6° ≈0.5-
(1)𝑖𝐴𝐵= 1:
40
9
(2)üǏƸǐpŽüŨĝƸƹ(Yoě
* NOPQR§]\]>HU=>HXSTUVWXY`ab@(ymÎÏU
Ǒǒ§]\]>gHh
#1-oĀs
∠𝐷𝐶𝐸= 26.6°(∠𝐴𝐷𝐸= ∠𝐸= 90°(:;
@𝐶𝐶𝐼⊥𝐴𝐷@𝐼(Ǔ~G>𝐶𝐼𝐷𝐸g=
>(K𝐶𝐸= 𝐼𝐷U𝐶𝐼= 𝐷𝐸= 1.8 mtwx<Rt △𝐶𝐷𝐸¤Rt △𝐴𝐼𝐶?§]\]>vrt
#2-oĀs
∠𝑃𝐹𝐴= 45°(n<Rt △𝑃𝐴𝐹?§]\]>rs𝐴𝐹= 16.4 m(:;
@𝐹𝐹𝐻⊥𝐴𝐵
@𝐻(klVWXY¤§]\]>rs𝐹𝐻2 +𝐴𝐻2 = 𝐴𝐹2(u𝐹𝐻= 9𝑥(K𝐴𝐻= 40𝑥(K(9𝑥)2 + (40𝑥)2 =
16.42(s𝑥=
2
5(pqs𝐹𝐻(ŭx 3 DžÂǔvr
}#1-
oĀs
∠𝐷𝐶𝐸= 26.6°(∠𝐴𝐷𝐸= ∠𝐸= 90°
:;
@𝐶𝐶𝐼⊥𝐴𝐷@𝐼(
∴∠𝐶𝐼𝐷= 90°
∴~G>𝐶𝐼𝐷𝐸g=>
∴𝐶𝐸= 𝐼𝐷(𝐶𝐼= 𝐷𝐸= 1.8 m
<Rt △𝐶𝐷𝐸?(∠𝐸= 90°(𝐷𝐸= 1.8 m
∴tan∠𝐷𝐶𝐸=
𝐷𝐸
𝐶𝐸= 0.5 s
𝐶𝐸= 𝐼𝐷= 3.6 m
∴𝐴𝐼= 8m
<Rt △𝐴𝐼𝐶?(∠𝐴𝐼𝐶= 90∘(
∴tan∠𝐶𝐴𝐷=
𝐼𝐶
𝐴𝐼=
1.8
8 =
9
40(
∴𝑖𝐴𝐵= 1:
40
9
#2-
oĀs
∠𝑃𝐹𝐴= 45°
<Rt △𝑃𝐴𝐹?(∠𝑃𝐴𝐹= 90°(𝑃𝐴= 16.4 m(tan∠𝑃𝐹𝐴=
𝑃𝐴
𝐴𝐹= 1(
∴𝐴𝐹= 16.4 m(
:;
@𝐹𝐹𝐻⊥𝐴𝐵@𝐻(
<Rt △𝐴𝐻𝐹?(∠𝐴𝐻𝐹= 90°(tan∠𝐹𝐴𝐻=
𝐹𝐻
𝐴𝐻=
9
40(𝐹𝐻2 +𝐴𝐻2 = 𝐴𝐹2(
u𝐹𝐻= 9𝑥(K𝐴𝐻= 40𝑥(
∴(9𝑥)2 + (40𝑥)2 = 16.42(s𝑥=
2
5
∴𝐹𝐻= 3.6 m(
∵3𝑚< 3.6m vüǏƸǐpŽüŨĝƸƹ
24 #12 *-#24-25 ./01·ĿǕ¥ǖ·6789-:;(<Rt △𝐴𝐵𝐶?(∠𝐴𝐶𝐵= 90°(𝐴𝐵= 5cm(
tan∠𝐴𝐵𝐶=
3
4(𝐴𝐷∥𝐵𝐶(Ó𝐴𝐷= 2𝐵𝐶 ò@ P ú@ B ķZHǗÔB7𝐵𝐷ďǘ@ D ǙǗªò(1 ǚx
ò@ Q ú@ D ķZ2cm/sHǗÔB7𝐷𝐴ďǘ@ A ǙǗªò(u@ P ªòH÷ǛM t#s-
(1)@ Q ķx(u~G>𝐴𝐶𝑃𝑄H½¾MS(cm2)( S t H^_żǜ¿t
(2)ĕ𝑃𝑄⊥𝐴𝐵÷( t HLt
(3)JZ𝑄𝐷U𝑄𝑃MG▱𝐷𝑄𝑃𝐸(<Ėõªòó?(gơƢ<ƣ ÷Ư t(E@ E <∠𝐷𝐴𝐵H*1ƤJ
Ƣ<( t HLtJŬƢ<(ƥĵYo
(1)𝑆=
3
5𝑡2−
24
5 𝑡+ 21
(2)𝑡=
75
47
(3)Ƣ<(𝑡=
35
23
* #1-®Ò𝐴𝑃(@𝑃𝑃𝑀⊥𝐴𝐶𝑀(𝑃𝑁⊥𝐴𝐷𝑁(@𝐵𝐵𝐻⊥𝑃𝑀𝐻(K~G>𝐵𝐶𝑀𝐻U~
G>𝐴𝑀𝑃𝑁ƭg=>(s𝑃𝑁= 𝐴𝐶−𝐵𝐻(𝐴𝑁= 𝑀𝑃(𝑀𝐻= 𝐵𝐶= 4(𝑃𝑀∥𝐴𝐷(K
∠𝐵𝑃𝐻= ∠𝐵𝐷𝐴= ∠𝐴𝐵𝐶(𝐵𝐻=
3
5𝑡(𝑃𝐻=
4
5𝑡(𝑃𝑀= 4 +
4
5𝑡(𝑃𝑁= 3−
3
5𝑡(oĀs𝐷𝑄= 2(𝑡−1)(Ó
1 < 𝑡≤5(K𝐴𝑄= 8−2(𝑡−1)(o𝑆= 𝑆Δ𝐴𝑃𝐶+ 𝑆Δ𝐴𝑃𝑄=
1
2𝐴𝐶⋅𝑃𝑀+
1
2𝐴𝑄⋅𝑃𝑁(vrs¯Şt
#2-𝑃𝑄⊥𝐴𝐵÷(
𝑄𝑁
𝑃𝑁=
3
4(¶Ž𝑄𝑁U𝑃𝑁HLvrs¯Şt
#3-𝐸𝑃𝐴𝐵𝐿(@𝐵𝐵𝑆⊥𝑃𝐿𝑆(𝐵𝑃= 𝐵𝐿(𝑃𝑆= 𝐿𝑆(𝐴𝐿= 𝐸𝐿(s5−𝑡=
8
5𝑡+ 2(𝑡−1)(
vrs¯Ş
}#1-
®Ò𝐴𝑃(@𝑃𝑃𝑀⊥𝐴𝐶𝑀(𝑃𝑁⊥𝐴𝐷𝑁(@𝐵𝐵𝐻⊥𝑃𝑀𝐻(:;
K~G>𝐵𝐶𝑀𝐻U~G>𝐴𝑀𝑃𝑁ƭg=>(
∴𝑃𝑁= 𝐴𝐶−𝐵𝐻(𝐴𝑁= 𝑀𝑃(𝑀𝐻= 𝐵𝐶= 4(𝑃𝑀∥𝐴𝐷(
∴∠𝐵𝑃𝐻= ∠𝐵𝐷𝐴= ∠𝐴𝐵𝐶(
∴sin∠𝐵𝑃𝐻=
𝐵𝐻
𝐵𝑃= sin∠𝐴𝐵𝐶=
𝐴𝐶
𝐴𝐵=
3
5(
∴𝐵𝐻=
3
5𝐵𝑃=
3
5𝑡(
cos∠𝐵𝑃𝐻=
𝑃𝐻
𝐵𝑃= cos∠𝐴𝐵𝐶=
𝐵𝐶
𝐴𝐵=
4
5(
∴𝑃𝐻=
4
5𝐵𝑃=
4
5𝑡(
∴𝑃𝑀= 𝑀𝐻+ 𝑃𝐻= 4 +
4
5𝑡(𝑃𝑁= 𝐴𝐶−𝐵𝐻= 3−
3
5𝑡(
oĀs
𝐷𝑄= 2(𝑡−1)(Ó1 < 𝑡≤5(
∴𝐴𝑄= 𝐴𝐷−𝐷𝑄= 8−2(𝑡−1)(
∴𝑆= 𝑆△𝐴𝑃𝐶+ 𝑆△𝐴𝑃𝑄= 1
2 𝐴𝐶⋅𝑃𝑀+ 1
2 𝐴𝑄⋅𝑃𝑁
= 1
2 × 3 × 4 +
4
5 𝑡+ 1
2 × [8−2(𝑡−1)] × 3−
3
5 𝑡
=
3
5𝑡2−
24
5 𝑡+ 21t
#2-
J𝑃𝑄⊥𝐴𝐵(K∠𝑃𝑄𝑁+ ∠𝐵𝐴𝑄= 90°(
∵∠𝑃𝑄𝑁+ ∠𝑄𝑃𝑁= 90°(
∴∠𝑄𝑃𝑁= ∠𝐵𝐴𝑄= ∠𝐴𝐵𝐶(
∴tan∠𝑄𝑃𝑁=
𝑄𝑁
𝑃𝑁= tan∠𝐴𝐵𝐶=
𝐴𝐶
𝐵𝐶=
3
4(
∵𝑄𝑁= 𝐴𝐷−𝐴𝑁−𝐷𝑄= 8−4 +
4
5 𝑡−2(𝑡−1) = 6−
14
5 𝑡(𝑃𝑁= 3−
3
5𝑡(
∴
6−
14
5 𝑡
3−
3
5 𝑡=
3
4(
s
𝑡=
75
47t
#3-
Ƣ<(𝐸𝑃𝐴𝐵𝐿(@𝐵𝐵𝑆⊥𝑃𝐿𝑆(:;
∵𝐴𝐸*∠𝐷𝐴𝐵(
∴∠𝐷𝐴𝐸= ∠𝐵𝐴𝐸(
∵~G>𝐷𝑄𝑃𝐸g~G>(
∴𝑃𝐸= 𝐷𝑄(𝐴𝐷∥𝐸𝐿(
∴∠𝐿𝑃𝐵= ∠𝐴𝐷𝐵= ∠𝐴𝐵𝐶(∠𝐵𝑃𝐿= ∠𝐵𝐷𝐴= ∠𝐵𝐴𝐷= ∠𝐵𝐿𝑃(∠𝐷𝐴𝐸= ∠𝐴𝐸𝐿(
∴𝐵𝑃= 𝐵𝐿(∠𝐵𝐴𝐸= ∠𝐴𝐸𝐿(
∴𝑃𝑆= 𝐿𝑆(𝐴𝐿= 𝐸𝐿(
𝐴𝐿= 𝐴𝐵−𝐵𝐿= 5−𝐵𝑃= 5−𝑡(
cos∠𝐿𝑃𝐵=
𝑃𝑆
𝐵𝑃= cos∠𝐴𝐵𝐶=
𝐵𝐶
𝐴𝐵=
4
5(
∴𝑃𝑆=
4
5𝐵𝑃=
4
5𝑡(
∴𝑃𝐿= 2𝑃𝑆=
8
5𝑡(
∴𝐸𝐿= 𝑃𝐿+ 𝑃𝐸= 𝑃𝐿+ 𝐷𝑄=
8
5𝑡+ 2(𝑡−1)(
∴5−𝑡=
8
5𝑡+ 2(𝑡−1)(
s
𝑡=
35
23
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