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\(\dfrac{a^2+b^2}{b^2+c^2}\)
\(=\dfrac{a^2+ac}{ac+c^2}\)(vì b2=ac)
\(=\dfrac{a\left(a+c\right)}{c\left(a+c\right)}\)(đặt a,c ra ngoài)
\(=\dfrac{a}{c}\)(rút gọn a+c)
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Tham khảo:Chứng minh a/b=c/d hoặc a/b=d/c biết (a^2+b^2)/(c^2+d^2)=ab/cd - An Nhiên
\(\text{Cho }\dfrac{a}{b}=\dfrac{d}{c}\text{ và }b,d\notin0\text{.CMR:}\dfrac{ac}{bd}=\dfrac{a^2+c^2}{b^2+d^2}\)
\(\text{Ta có:}\dfrac{a}{b}=\dfrac{c}{d}=k\)
\(\text{Lại có:}\dfrac{ac}{bd}=\dfrac{bk.dk}{bd}=\dfrac{\left(bd\right).k^2}{bd}=k^2\)
\(\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{\left(bk\right)^2+\left(dk\right)^2}{b^2+d^2}=\dfrac{b^2.k^2+d^2.k^2}{b^2+d^2}=\dfrac{\left(b^2+d^2\right).k^2}{b^2+d^2}=k^2\)
\(\Rightarrow\dfrac{ac}{bd}=\dfrac{a^2+c^2}{b^2+d^2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)
\(\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)
Suy ra: \(VT=\dfrac{bk^2\left(b+d\right)}{dk^2\left(d-b\right)}=\dfrac{b\left(b+d\right)}{d\left(d-b\right)}\)
\(VP=\dfrac{b^2+bd}{d^2-bd}=\dfrac{b\left(b+d\right)}{d\left(d-b\right)}\)
\(\Rightarrow VT=VP\rightarrowđpcm.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\Rightarrow a=bk;c=dk\)
Ta có: \(\dfrac{a^2+ac}{c^2-ac}=\dfrac{b^2k^2+bk\cdot dk}{d^2k^2-bk\cdot dk}=\dfrac{bk^2\cdot\left(b+d\right)}{dk^2\cdot\left(d-b\right)}=\dfrac{b\left(b+d\right)}{d\left(d-b\right)}\left(1\right)\)
\(\dfrac{b^2+bd}{d^2-bd}=\dfrac{b\left(b+d\right)}{d\left(d-b\right)}\left(2\right)\)
Từ (1) và (2) \(\Rightarrow\dfrac{a^2+ac}{c^2-ac}=\dfrac{b^2+bd}{d^2-bd}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\). \(\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)
Ta có:
\(\dfrac{ac}{bd}=\dfrac{bk.dk}{bd}=\dfrac{bdk^2}{bd}=k^2\)
\(\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{\left(bk\right)^2+\left(dk\right)^2}{b^2+d^2}=\dfrac{b^2k^2+d^2k^2}{b^2+d^2}=\dfrac{k^2\left(b^2+d^2\right)}{b^2+d^2}=k^2\)
\(\Rightarrow\dfrac{ac}{bd}=\dfrac{a^2+c^2}{b^2+d^2}\left(=k^2\right)\)
\(\Rightarrowđpcm\)
![](https://rs.olm.vn/images/avt/0.png?1311)
đặt a/b=c/d=k
=>a=bk;c=dk rồi cứ thế thay lần lượt vào ac/bd;a^2+c^2/b^2+d^2
full hd :))
Đặt:
\(\dfrac{a}{b}=\dfrac{c}{d}=t\Leftrightarrow\left\{{}\begin{matrix}a=bt\\c=dt\end{matrix}\right.\)
Khi đó:
\(\dfrac{ac}{bd}=\dfrac{bt.dt}{bd}=\dfrac{t^2bd}{bd}=t^2\)
\(\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{b^2t^2+d^2t^2}{b^2+d^2}=\dfrac{t^2\left(b^2+d^2\right)}{b^2+d^2}=t^2\)
Vậy.....
![](https://rs.olm.vn/images/avt/0.png?1311)
b^2=ac
=>b/a=c/b=k
=>b=ak; c=bk=ak*k=ak^2
\(\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{a^2+a^2k^2}{a^2k^2+a^2k^4}=\dfrac{1}{k^2}\)
\(\dfrac{a}{c}=\dfrac{a}{ak^2}=\dfrac{1}{k^2}\)
=>\(\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{a}{c}\)
Ai giúp đi, mik lazy lắm
\(\dfrac{a}{c}=\dfrac{a^2+b^2}{b^2+c^2}\)
\(VP=\dfrac{a^2+ac}{ac+c^2}=\dfrac{a\left(a+c\right)}{c\left(a+c\right)}=\dfrac{a}{c}=VT\left(đpcm\right)\)