\(Ay\) là tia phân giác trong của tam giác \(BAC\). 

Khi đó \(Ax\) và \(Ay\) vuông góc với nhau suy ra \(Ay\perp BE\).

Khi đó tam giác \(ABE\) có \(Ay\) là đường phân giác đồng thời là đường cao suy ra tam giác \(ABE\) cân tại \(A\) suy ra \(\widehat{ABE}=\widehat{AEB}\).

\(BH\) vuông góc với \(AD\) mà \(BE//AD\) suy ra \(BH\) vuông góc với \(BE\).

\(AI\) song song \(BH\) mà \(BH\) vuông góc với \(AD\) suy ra \(AI\) vuông góc với \(AD\)

suy ra \(AI\) là phân giác trong tam giác \(BAC\).

\(\dfrac{1}{9}-0,3. \dfrac{3}{9}+\dfrac{1}{3}\)

\(=\dfrac{1}{9}-\dfrac{1}{10}+\dfrac{1}{3}\)

\(=\dfrac{1}{90}+\dfrac{1}{3}=\dfrac{31}{90}\)

Lời giải:
$\frac{1}{9}-0,3.\frac{3}{9}+\frac{1}{3}=\frac{1}{9}-\frac{1}{10}+\frac{1}{3}=\frac{31}{90}$

`8/9 - 1/72 - 1/56 - 1/42 - 1/30 - 1/20 - 1/12 - 1/6 - 1/2`

`=8/9 - ( 1/2 - 1/6-1/12-1/20-1/30-1/42 - 1/56-1/72)`

`=8/9 - (1/(1xx2) + 1/(2xx3) + ...... +1/(8xx9))`

`=8/9 - ( 1-1/2+1/2-1/3+.....+1/8 -1/9)`

`=8/9 - (1-1/9)`

`=8/9 - (9/9-1/9)`

`=8/9 - 8/9`

`=0`

`#H.J`

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`0,5 + 1/3 + 0,4 + 5/7 + 1/6 - 4/35`

`=1/2 + 1/3 + 2/5 + 5/7 + 1/6 - 4/35`

`=(1/2 + 1/3 + 1/6) + ( 2/5 - 4/35 + 5/7)`

`= (3/6 + 2/6 + 1/6) + ( 14/35 - 4/35 + 25/35)`

`=((3+2+1)/6) + (( 14-4+25)/35)`

`= 6/6+ 35/35`

`=1+1`

`=2`

\(0,5+\dfrac{1}{3}+0,4+\dfrac{5}{7}+\dfrac{1}{6}-\dfrac{4}{35}\)

\(=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{2}{5}+\dfrac{5}{7}+\dfrac{1}{6}-\dfrac{4}{35}\)

\(=\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}\right)+\left(\dfrac{5}{7}+\dfrac{2}{5}-\dfrac{4}{35}\right)\)

\(=1+\dfrac{39}{35}-\dfrac{4}{35}\)

\(=1+1=2\)

`3.x^2 - 18= 0 `

`3.x^2 = 0+18`

`3.x^2=18`

`x^2=18:3`

`x^2=9`

`x^2=3^2`

`=> x=2`

3x^2-18=0

=>x^2=6

=>x=\(+-\sqrt{6}\)

\(\left|2x-\dfrac{1}{2}\right|=\dfrac{17}{4}\)

\(\left[{}\begin{matrix}2x=\dfrac{17}{4}+\dfrac{1}{2}=\dfrac{19}{4}\\2x=-\dfrac{17}{4}+\dfrac{1}{2}=-\dfrac{15}{4}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{19}{4}:2=\dfrac{19}{8}\\x=-\dfrac{15}{8}\end{matrix}\right.\)

\(3,5-|2x-\dfrac{1}{2}|=-0,75\)

\(|2x-\dfrac{1}{2}|=3,5-(-0,75)=4,25\)

\(\bullet TH1: 2x-\dfrac{1}{2}=4,25\)

     \(=>2x=4,25+\dfrac{1}{2}=4,75\)

    \(=>x=4,75:2=2,375\)

\(\bullet TH2:2x-\dfrac{1}{2}=-4,75\)

    \(=>2x=-4,75+\dfrac{1}{2}=-4,25\)

   \(=>x=-4,25:2=-2,125\)

`2/3x - 1/2 = (-2/3)^2`

`2/3x - 1/2= 4/9`

`2/3x = 4/9+1/2`

`2/3x=17/18`

`x=17/18:2/3`

`x=17/18 xx 3/2`

`x=17/12`

\(\dfrac{2}{3}x-\dfrac{1}{2}=\left(-\dfrac{2}{3}\right)^2\)

\(\dfrac{2}{3}x-\dfrac{1}{2}=\dfrac{4}{9}\)

\(\dfrac{2}{3}x=\dfrac{4}{9}+\dfrac{1}{2}\)

\(\dfrac{2}{3}x=\dfrac{17}{18}\)

x\(=\dfrac{17}{18}:\dfrac{2}{3}\)

x= \(\dfrac{17}{12}\)