Bài 5 hình 1: (tự vẽ hình nhé bạn)
a) Xét ΔABD và ΔACB ta có:
\(\widehat{BAD}\)= \(\widehat{BAC}\) (góc chung)
\(\widehat{ABD}\)= \(\widehat{ACB}\) (gt)
=> ΔABD ~ ΔACB (g-g)
=> \(\dfrac{AB}{AC}\) = \(\dfrac{BD}{CB}\) = \(\dfrac{AD}{AB}\) (tsđd)
b) Ta có: \(\dfrac{AB}{AC}\) = \(\dfrac{AD}{AB}\) (cm a)
=> \(AB^2\) = AD.AC
=> \(2^2\) = AD.4
=> AD = 1 (cm)
Ta có: AC = AD + DC (D thuộc AC)
      => 4   =   1   + DC
      => DC = 3 (cm)
c) Xét ΔABH và ΔADE ta có: 
   \(\widehat{AHB}\) = \(\widehat{AED}\) (=\(90^0\))
   \(\widehat{ADB}\) = \(\widehat{ABH}\) (ΔABD ~ ΔACB)
=> ΔABH ~ ΔADE
=> \(\dfrac{AB}{AD}\) = \(\dfrac{AH}{AE}\) = \(\dfrac{BH}{DE}\) (tsdd)
Ta có: \(\dfrac{S_{ABH}}{S_{ADE}}\) = \(\left(\dfrac{AB}{AD}\right)^2\)= \(\left(\dfrac{2}{1}\right)^2\)= 4
=> đpcm

Tiếp bài 5 hình 2 (tự vẽ hình)
a) Xét ΔABC vuông tại A ta có:
\(BC^2\) = \(AB^2\) + \(AC^2\)
\(BC^2\) = \(21^2\) + \(28^2\)
BC = 35 (cm)
b) Xét ΔABC và ΔHBA ta có:
\(\widehat{BAC}\) = \(\widehat{AHB}\) ( =\(90^0\))
\(\widehat{ABC}\) = \(\widehat{ABH}\) (góc chung)
=> ΔABC ~ ΔHBA (g-g)
=> \(\dfrac{AB}{BH}\) = \(\dfrac{BC}{AB}\) (tsdd)
=> \(AB^2\) = BH.BC
=> \(21^2\) = 35.BH
=> BH = 12,6 (cm)
c) Xét ΔABC ta có:
BD là đường p/g (gt)
=> \(\dfrac{AD}{DC}\) = \(\dfrac{AB}{BC}\) (t/c đường p/g)
Xét ΔABH ta có: 
BE là đường p/g (gt)
=> \(\dfrac{HE}{AE}\) = \(\dfrac{BH}{AB}\) (t/c đường p/g)
Mà: \(\dfrac{AB}{BC}\) = \(\dfrac{BH}{AB}\) (cm b)
=> đpcm
d) Ta có: \(\left\{{}\begin{matrix}\widehat{HBE}+\widehat{BEH}=90^0\\\widehat{ABD}+\widehat{ADB=90^0}\\\widehat{HBE}=\widehat{ABD}\end{matrix}\right.\)
=> \(\widehat{BEH}=\widehat{ADB}\)
Mà \(\widehat{BEH}=\widehat{AED}\) (2 góc dd)
Nên \(\widehat{ADB}=\widehat{AED}\)
=> đpcm

1 A

2 B

3 A

4 A

5 A

6 C

7 D

8 B

Bài 1

a) \(\dfrac{3}{8}+\dfrac{5}{8}=\dfrac{8}{8}=1\)             

b) \(\dfrac{3}{4}+\dfrac{-7}{16}=\dfrac{12}{16}+\dfrac{-7}{16}=\dfrac{5}{16}\)

c) \(2\dfrac{17}{20}-\dfrac{1}{2}+3\dfrac{3}{20}=\dfrac{57}{20}-\dfrac{1}{2}+\dfrac{63}{20}\)\(=\dfrac{47}{20}+\dfrac{63}{20}=\dfrac{110}{20}=\dfrac{11}{2}\)

d) \(\dfrac{2}{3}-2\dfrac{1}{8}+\dfrac{7}{24}=\dfrac{2}{3}-\dfrac{17}{8}+\dfrac{7}{24}=\dfrac{16}{24}-\dfrac{51}{24}+\dfrac{7}{24}=\dfrac{16-51+7}{24}=\dfrac{-28}{24}=\dfrac{-7}{6}\)

Bài 2 :

a) \(x-\dfrac{7}{4}=3\)

\(x=3+\dfrac{7}{4}\)

\(x=\dfrac{19}{4}\)

b) \(x-\dfrac{1}{2}=\dfrac{4}{16}\cdot\dfrac{8}{3}\)

   \(x-\dfrac{1}{2}=\dfrac{2}{3}\)

  \(x=\dfrac{2}{3}+\dfrac{1}{2}\)

 \(x=\dfrac{5}{6}\)

c) \(\dfrac{15}{11}\div x=\dfrac{45}{22}\)

             \(x=\dfrac{15}{11}\div\dfrac{45}{22}\)

            \(x=\dfrac{2}{3}\)

d) \(\dfrac{8}{3}-2x=\dfrac{8}{5}-1\)

    \(\dfrac{8}{3}-2x=\dfrac{3}{5}\)

           \(2x=\dfrac{8}{3}-\dfrac{3}{5}\)

           \(2x=\dfrac{31}{15}\)

            \(x=\dfrac{31}{15}\div2\)

           \(x=\dfrac{31}{30}\)

1)x>=0

2)v8+5v2-v32+6v1/2=2v2+5v2-4v2+3v2=9v2

3)vx+1=2

x+1=4=>x=3

câu hình:

a) Vì C là điểm chính giữa cung AB \(\Rightarrow OC\bot AB\Rightarrow\angle AOC=90\)

\(\Rightarrow\angle AOC=\angle AHC\Rightarrow AOHC\) nội tiếp

b) Vì AOHC nội tiếp \(\Rightarrow\angle CHO=180-\angle CAO=180-\angle CAB=\angle CNB\)(CANB nội tiếp)

c) Xét \(\Delta CHM\) và \(\Delta ACM:\) Ta có: \(\left\{{}\begin{matrix}\angle CHM=\angle ACM=90\\\angle CMAchung\end{matrix}\right.\)

\(\Rightarrow\Delta CHM\sim\Delta ACM\left(g-g\right)\Rightarrow\dfrac{HM}{CM}=\dfrac{CM}{MA}\)

Xét \(\Delta BNM\) và \(\Delta ACM:\) Ta có: \(\left\{{}\begin{matrix}\angle BMN=\angle AMC\\\angle CAM=\angle MBN\left(ACNBnt\right)\end{matrix}\right.\)

\(\Rightarrow\Delta BNM\sim\Delta ACM\left(g-g\right)\Rightarrow\dfrac{MN}{BM}=\dfrac{CM}{MA}\)

\(\Rightarrow\dfrac{MN}{BM}=\dfrac{MH}{CM}\) mà \(BM=CM\Rightarrow MH=MN\)

\(\Rightarrow BHCN\) là hình bình hành (2 đường chéo giao nhau tại trung điểm mỗi đường)

\(\Rightarrow\angle IHB=\angle ICN=90-\angle CNA=90-\angle CBA=45\) (C là điểm chính giữa)

mà \(\angle IHO=\angle CAO=45\Rightarrow\angle OHB=90\Rightarrow OH\bot HB\)

Ta có: \(CH^2=AH.HM\Rightarrow AH=\dfrac{CH^2}{HM}=\dfrac{NB^2}{\dfrac{1}{2}HN}=\dfrac{2BN^2}{HN}\)

Lại có: \(\angle NHB=90-\angle BHI=90-45=45\Rightarrow\Delta NHB\) vuông cân

\(\Rightarrow BN=HN\Rightarrow AH=\dfrac{2BN^2}{BN}=2BN=BN+HN\)

d) Vì \(\angle OHI=\angle BHI=45\Rightarrow HI\) là phân giác \(\angle OHB\)

\(\Rightarrow\dfrac{IO}{IB}=\dfrac{OH}{HB}\)

Xét \(\Delta OHB\) và \(\Delta CHA:\) Ta có: \(\left\{{}\begin{matrix}\angle CHA=\angle OHB=90\\\angle ACH=\angle HOB\end{matrix}\right.\)

\(\Rightarrow\Delta OHB\sim\Delta CHA\left(g-g\right)\Rightarrow\dfrac{OH}{HB}=\dfrac{CH}{AH}=\dfrac{BN}{BN+HN}=\dfrac{BN}{2BN}=\dfrac{1}{2}\)

\(\Rightarrow\dfrac{IO}{IB}=\dfrac{1}{2}\Rightarrow IB=2IO\)

câu 5 ta có: \(2021\left(x^2+y^2+z^2\right)=3xyz\)

\(=>\dfrac{x^2+y^2+z^2}{xyz}=\dfrac{3}{2021}< =>\dfrac{x}{yz}+\dfrac{y}{xz}+\dfrac{z}{xy}=\dfrac{3}{2021}\)

Áp dụng BDT Cô si 

\(=>\left\{{}\begin{matrix}\dfrac{x}{yz}+\dfrac{y}{xz}\ge\dfrac{2}{z}\\\dfrac{y}{xz}+\dfrac{z}{xy}\ge\dfrac{2}{x}\\\dfrac{x}{yz}+\dfrac{z}{xy}\ge\dfrac{2}{y}\end{matrix}\right.\)\(\)

\(=>\left(\dfrac{x}{yz}+\dfrac{y}{xz}\right)+\left(\dfrac{y}{xz}+\dfrac{z}{xy}\right)+\left(\dfrac{x}{yz}+\dfrac{z}{xy}\right)\ge2\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)

\(=>\dfrac{x}{yz}+\dfrac{y}{xz}+\dfrac{z}{xy}\ge\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\)

\(=>\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\le\dfrac{3}{2021}\)

Áp dụng cố si \(=>x^2+yz\ge2x\sqrt{yz}=>\dfrac{x}{x^2+yz}\le\dfrac{1}{2\sqrt{yz}}=\dfrac{1}{4}.2.\dfrac{1}{\sqrt{y}}.\dfrac{1}{\sqrt{z}}\)\(=\dfrac{1}{4}\left(\dfrac{1}{y}+\dfrac{1}{z}\right)\)(1)

tương tự \(=>\dfrac{y}{y^2+zx}\le\dfrac{1}{4}\left(\dfrac{1}{z}+\dfrac{1}{x}\right)\left(2\right)\)

\(\dfrac{z}{z^2+xy}\le\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\left(3\right)\)

cộng vế (1)(2)(3)

\(=>A\le\dfrac{1}{4}\left[\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{x}+\dfrac{1}{z}+\dfrac{1}{y}\right]\)\(=\dfrac{1}{4}.2\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)\)

\(=\dfrac{1}{2}.\dfrac{3}{2021}=\dfrac{3}{4042}\). Dấu"=" xảy ra<=>\(x=y=z=\dfrac{1}{2021}\)

vậy Max \(=\dfrac{3}{4042}\)

- Trích mẫu thử.

- Cho quỳ tím vào các mẫu thử:

+ Quỳ tím hóa đỏ: HCl, H₂SO₄(1)

+ Quỳ tím ko đổi màu: NaNO₃

- Cho BaCl₂ vào nhóm (1)

+ Xuất hiện kết tủa: H₂SO₄

+ Ko kết tủa: HCl

PTHH: \(BaCl_2+H_2SO_4\rightarrow2HCl+BaSO_4\downarrow\)

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