Bài 1:

Áp dụng HTL trong tam giác vuông:
$AH^2=BH.CH$

$\Leftrightarrow x^2=4.9=36$

$\Rightarrow x=6$ (do $x>0$)

Bài 2:

Áp dụng định lý Pitago:

$BC=\sqrt{AB^2+AC^2}=\sqrt{6^2+8^2}=10$ (cm)

$\sin B=\frac{AC}{BC}=\frac{6}{10}=\frac{3}{5}$

$\Rightarrow \widehat{B}=36,87^0$

$\widehat{C}=90^0-\widehat{B}=90^0-36,87^0=53,13^0$

Bài 3:

a. Áp dụng định lý Pitago:

$BC=\sqrt{AB^2+AC^2}=\sqrt{3^2+4^2}=5$ (cm)

$AH=\frac{2S_{ABC}}{BC}=\frac{AB.AC}{BC}=\frac{3.4}{5}=2,4$ (cm)

b.

$\sin B=\frac{AC}{BC}=\frac{4}{5}$

$\Rightarrow \widehat{B}=53,13^0$

$\widehat{C}=90^0-\widehat{B}=36,87^0$

c.

Áp dụng tính chất tia phân giác:

$\frac{BE}{CE}=\frac{AB}{AC}=\frac{3}{4}$

$\Rightarrow \frac{BE}{BC}=\frac{3}{7}$

$\Rightarrow BE=BC.\frac{3}{7}=\frac{5.3}{7}=\frac{15}{7}$  (cm)

$CE=BC-BE=5-\frac{15}{7}=\frac{20}{7}$ (cm)

a) \(\dfrac{2x}{3}\)+\(\dfrac{2x-1}{6}\)=4 - \(\dfrac{x}{3}\)

<=>\(\dfrac{2x}{3}\)+\(\dfrac{2x-1}{6}\) - 4+\(\dfrac{x}{3}\)=0

<=>\(\dfrac{2x.2+2x-1-4.6+x.2}{6}\)=0

=>4x-2x-24+2x=0

<=>4x-24=0

<=>4x=24

<=>x=6

Vậy x=6

b)\(\dfrac{x-1}{2}\)+\(\dfrac{x-1}{4}\)=1 - \(\dfrac{2\left(x-1\right)}{3}\)

<=>\(\dfrac{x-1}{2}\)+\(\dfrac{x-1}{4}\)-1+\(\dfrac{2\left(x-1\right)}{3}\)=0

<=>\(\dfrac{6.\left(x-1\right)+3\left(x-1\right)-1.12+4.2\left(x-1\right)}{12}\)=0

=>6x-6+3x-3-12+4x-4+2x-2=0

<=>15x-27=0

<=>15x=27

<=>x=\(\dfrac{9}{5}\)

Vậy x=\(\dfrac{9}{5}\)

d)\(\dfrac{7x}{8}\)-5(x-9)=20x+\(\dfrac{1,5}{6}\)

<=>\(\dfrac{7x}{8}\)-5(x-9)-20x-\(\dfrac{1,5}{6}\)=0

<=>\(\dfrac{3.7x-24.5\left(x-9\right)-20x.24-1,5.4}{24}\)=0

=>-478x+165=0

<=> - 478= - 165

<=> x=\(\dfrac{165}{487}\)

Vậy x =\(\dfrac{165}{487}\)

Lời giải.

c.

$x^3-3x^2+3x-1=0$

$\Leftrightarrow (x-1)^3=0$

$\Leftrightarrow x-1=0$

$\Leftrightarrow x=1$

Vậy pt có tập nghiệm $S=\left\{1\right\}$

d. ĐKXĐ: $x\neq \frac{-1}{3}; -3$

PT $\Leftrightarrow \frac{(3x-1)(x+3)+(x-3)(3x+1)}{(3x+1)(x+3)}=2$

$\Leftrightarrow \frac{6x^2-6}{3x^2+10x+3}=2$

$\Leftrightarrow 6x^2-6=2(3x^2+10x+3)$

$\Leftrightarrow 20x+12=0$

$\Leftrightarrow x=\frac{-3}{5}$ (tm)

Vậy tập nghiệm của pt là $S=\left\{\frac{-3}{5}\right\}$

Bài 2:

a. 

\(\left\{\begin{matrix} 2x-3y=11\\ 5x-4y=3\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} 10x-15y=55\\ 10x-8y=6\end{matrix}\right.\)

\(\Rightarrow (10x-8y)-(10x-15y)=6-55\)

\(\Leftrightarrow 7y=-49\Leftrightarrow y=-7\)

\(x=\frac{3y+11}{2}=\frac{3.(-7)+11}{2}=-5\)

Vậy hpt có nghiệm $(x,y)=(-5,-7)$

b. Không đủ cơ sở để tìm $x,y$

c. 

\(\left\{\begin{matrix} 5x+3y=\lambda\\ -x+\lambda y=-8\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} 5x+3y=\lambda\\ -5x+5\lambda y=-40\end{matrix}\right.\)

\(\Rightarrow (3+5\lambda)y=\lambda-40\)

Nếu $\lambda = \frac{-3}{5}$ thì $0.y=\frac{-203}{5}$ (vô lý) nên hpt vô nghiệm

Nếu $\lambda \neq \frac{-3}{5}$ thì:

$y=\frac{\lambda - 40}{3+5\lambda}$

$x=8+\lambda y=\frac{\lambda ^2+24}{5\lambda +3}$

Bài nào em ??

Các bài ở hình ảnh trên

Bài 1: 

a) Ta có: \(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{2020\cdot2021}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2020}-\dfrac{1}{2021}\)

\(=1-\dfrac{1}{2021}\)

\(=\dfrac{2020}{2021}\)

b) Ta có: \(B=\dfrac{3}{3\cdot5}+\dfrac{3}{5\cdot7}+...+\dfrac{3}{47\cdot49}\)

\(=\dfrac{3}{2}\left(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+\dfrac{2}{7\cdot9}+...+\dfrac{2}{47\cdot49}\right)\)

\(=\dfrac{3}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{47}-\dfrac{1}{49}\right)\)

\(=\dfrac{3}{2}\left(\dfrac{1}{3}-\dfrac{1}{49}\right)\)

\(=\dfrac{3}{2}\cdot\dfrac{46}{147}\)

\(=\dfrac{46}{2}\cdot\dfrac{3}{147}\)

\(=\dfrac{23}{49}\)

Bài 2: 

Ta có: \(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{99\cdot100}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}\)

\(=1-\dfrac{1}{100}\)

\(=\dfrac{99}{100}< 1\)