+ \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\)

\(\Rightarrow\frac{a}{a-b}=\frac{c}{c-d}\)

+ \(\frac{a}{c}=\frac{3a}{3c}=\frac{b}{d}=\frac{3a+b}{3c+d}\) \(\Rightarrow\frac{a}{3a+b}=\frac{c}{3c+d}\)

+ \(\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\Rightarrow\frac{a^2}{c^2}=\frac{\left(a-b\right)^2}{\left(c-d\right)^2}\)

\(\Rightarrow\frac{a\cdot b}{c\cdot d}=\frac{\left(a-b\right)^2}{\left(c-d\right)^2}\)

\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a^2}{b^2}=\frac{c^2}{d^2}=\frac{a^2+c^2}{b^2+d^2}\)

\(\Rightarrow\frac{a}{b}\cdot\frac{a}{b}=\frac{a^2+c^2}{b^2+d^2}\Rightarrow\frac{a\cdot c}{b\cdot d}=\frac{a^2+c^2}{b^2+d^2}\)

câu cuối lm tương tự

Từ \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{a}{c}\cdot\frac{a}{c}=\frac{a}{c}\cdot\frac{b}{d}=\frac{b}{d}\cdot\frac{b}{d}\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{ab}{cd}\)

a)Áp dụng t/c của dãy tỉ số bằng nhau ta có:

\(\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{ab}{cd}=\frac{a^2-b^2}{c^2-d^2}\)

b)\(\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{ab}{cd}\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{ab}{cd}=\frac{2ab}{2cd}\)

Áp dụng t/c của dãy tỉ số bằng nhau ta có:

\(\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{ab}{cd}=\frac{2ab}{2cd}=\frac{a^2+b^2-2ab}{c^2+d^2-2cd}=\frac{\left(a-b\right)^2}{\left(c-d\right)^2}\)

a) Ta có:\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\)

 => \(\left(\frac{a}{c}\right)^2=\left(\frac{b}{d}\right)^2\) =>  \(\frac{a^2}{c^2}=\frac{b^2}{d^2}\)

 Áp dụng t/c dãy tỉ số = nhau được: \(\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2-b^2}{c^2-d^2}\)

 Mặt khác, \(\frac{a^2}{c^2}=\frac{a}{c}.\frac{a}{c}=\frac{a}{c}.\frac{b}{d}=\frac{ab}{cd}\)

 Vậy \(\frac{a^2-b^2}{c^2-d^2}=\frac{ab}{cd}\left(=\left(\frac{a}{c}\right)^2\right)\)

b) \(\frac{a}{c}=\frac{b}{d}\)(câu a) => \(\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\) (t/c dãy tỉ số = nhau)

=> \(\left(\frac{a}{c}\right)^2=\frac{\left(a-b\right)^2}{\left(c-d\right)^2}\)

 Mặt khác, \(\left(\frac{a}{c}\right)^2=\frac{ab}{cd}\)(câu a) nên \(\frac{ab}{cd}=\frac{\left(a-b\right)^2}{\left(c-d\right)^2}=\left(\frac{a}{c}\right)^2\)

\(\frac{a}{b}=\frac{c}{d}\)\(\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}\)

\(\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2+b^2}{c^2+d^2}=\frac{a^2-b^2}{c^2-d^2}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\left(\frac{a+b}{c+d}\right)^2=\frac{ab}{cd}\)

Vậy \(\frac{ab}{cd}=\frac{a^2-b^2}{c^2-d^2}\)và \(\left(\frac{a+b}{c+d}\right)^2=\frac{a^2+b^2}{c^2+d^2}\)

Đặt Bằng a = bk 

c = dk Rồi thay vào biểu thức nha bạn

thank you

Đặt \(\frac{a}{b}=\frac{c}{d}=k\)

=>a=b.k

c=d.k

ta có Vế Phải : \(\frac{a^2+b^2}{c^2+d^2}=\frac{\left(b.k\right)^2+b^2}{\left(d.k\right)^2+d^2}=\frac{b^2.k^2+b^2}{d^2.k^2+d^2}=\frac{b^2.\left(k^2+1\right)}{d^2.\left(k^2+1\right)}=\frac{b^2}{d^2}\)

Vế Trái :\(\frac{ab}{cd}=\frac{b.k.b}{d.k.d}=\frac{b^2.k}{d^2.k}=\frac{b^2}{d^2}\)

vì \(\frac{b^2}{d^2}=\frac{b^2}{d^2}\)

=>VP=VT

=>\(\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}\)

Đặt \(\frac{a}{b}=\frac{c}{d}=k\)

=> a=b.k; c=d.k

Suy ra:

\(\frac{a^2+b^2}{c^2+d^2}=\frac{\left(b.k\right)^2+b^2}{\left(d.k\right)^2+d^2}=\frac{b^2}{d^2}\) (1)

\(\frac{ab}{cd}=\frac{b.k.b}{d.k.d}=\frac{b^2}{d^2}\) (2)

Từ (1) và (2) => \(\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}\)

Chúc bạn học tốt!

"." là nhân nha

Áp dụng tính chất của dãy tỉ số bằng nhau

a^2+b^2/c^2+d^2  =   a^2/c^2  =   b^2 / d^2

=>a/c   =    b/d

=>a/b    =    c/d

Chúc bạn học tốt nha

dat k ; ta co a= bk , c=dk , roi tu thay vao ma rut gon nhe

Ta có \(\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}=\frac{2ab}{2cd}\)

\(\Rightarrow\frac{a^2+b^2}{c^2+d^2}=\frac{2ab}{2cd}=\frac{a^2+b^2+2ab}{c^2+d^2+2cd}=\frac{a^2+b^2-2ab}{c^2+d^2-2cd}\)

\(\Rightarrow\frac{\left(a-b\right)^2}{\left(c-d\right)^2}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)

\(\Rightarrow\frac{a-b}{c-d}=\frac{a+b}{c+d}=\frac{a-b-a-b}{c-d-c-d}=\frac{a-b+a+b}{c-d+c+d}\)

\(\Rightarrow\frac{2b}{2d}=\frac{2a}{2c}\Rightarrow\frac{a}{b}=\frac{c}{d}\)

1, \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{3a}{3c}=\frac{b}{d}=\frac{3a+b}{3c+d}\Rightarrow\frac{a}{c}=\frac{3a+b}{3c+d}\Rightarrow\frac{a}{3a+b}=\frac{c}{3c+d}\)

2, a, Ta có: \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{a}{c}\cdot\frac{a}{c}=\frac{a}{c}\cdot\frac{b}{d}\Rightarrow\frac{a^2}{c^2}=\frac{ab}{cd}\)

\(\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{a}{c}\cdot\frac{b}{d}=\frac{b}{d}\cdot\frac{b}{d}\Rightarrow\frac{ab}{cd}=\frac{b^2}{d^2}\)

\(\Rightarrow\frac{ab}{cd}=\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2-b^2}{c^2-d^2}\)

b, Ta có: \(\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\Rightarrow\frac{a}{c}\cdot\frac{b}{d}=\frac{a-b}{c-d}\cdot\frac{a-b}{c-d}\Rightarrow\frac{ab}{cd}=\frac{\left(a-b\right)^2}{\left(c-d\right)^2}\)