Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Ta có: \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}\)
Áp dụng tính chất của dãy tỉ số = nhau ta có:
\(\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\left(\frac{a+b}{c+d}\right)^2\left(1\right)\)
\(\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2+b^2}{c^2+d^2}\left(2\right)\)
Từ (1) và (2) \(\Rightarrow\left(\frac{a+b}{c+d}\right)^2=\frac{a^2+b^2}{c^2+d^2}\left(đpcm\right)\)
đặt a/b = c/d = k (k thuộc N)
=> a = bk
c = dk
thay a và c vào 2 phân số cần so sánh thì = nhau
a)\(\frac{ab}{cd}=\frac{bk.b}{dk.b}=\frac{b^2}{d^2}\left(1\right)\)
\(\frac{a^2-b^2}{c^2-d^2}=\frac{b^2k^2-b^2}{d^2k^2-d^2}=\frac{b^2\left(k^2-1\right)}{d^2\left(k^2-1\right)}=\frac{b^2}{d^2}\left(2\right)\)
từ\(\left(1\right)\)và\(\left(2\right)\)\(\Rightarrow\frac{ab}{cd}=\frac{a^2-b^2}{c^2-d^2}\)
\(\frac{a}{b}=\frac{c}{d}\Rightarrow\hept{\begin{cases}a=bk\\c=dk\end{cases}}\)
ta có : \(\frac{4a-3b}{a}=\frac{4bk-3b}{bk}=\frac{b\left(4k-3\right)}{bk}=\frac{4k-3}{k}\)
\(\frac{4c-3d}{c}=\frac{4dk-3d}{dk}=\frac{d\left(4k-3\right)}{dk}=\frac{4k-3}{k}\)
\(\Rightarrow\frac{4a-3b}{a}=\frac{4c-3d}{c}\)
Đặt \(\frac{a}{b}=\frac{c}{d}=k\) thì \(a=bk,c=dk\).
\(\frac{2a+3b}{2a-3b}=\frac{2bk+3b}{2bk-3b}=\frac{b\left(2k+3\right)}{b\left(2k-3\right)}=\frac{2k+3}{2k-3}\\ \frac{2c+3d}{2c-3d}=\frac{2dk+3d}{2dk-3d}=\frac{d\left(2k+3\right)}{d\left(2k-3\right)}=\frac{2k+3}{2k-3}\)
Do đó: \(\frac{2a+3b}{2a-3b}=\frac{2c+3d}{2c-3d}\)
\(\frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+d}\)(tc dãy tỉ số bằng nhau)
\(\Rightarrow\frac{a}{b}\cdot\frac{c}{d}=\frac{a+c}{b+d}\cdot\frac{a+c}{b+d}\Rightarrow\frac{ac}{bd}=\frac{\left(a+c\right)^2}{\left(b+d\right)^2}\)
Ta có:
\(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=bk;c=dk\)
a) \(\frac{2a+3b}{2a-3b}=\frac{2bk+3b}{2bk-3b}=\frac{b\left(2k+3\right)}{b\left(2k-3\right)}=\frac{2k+3}{2k-3}\left(1\right)\)
\(\frac{2c+3d}{2c-3d}=\frac{2dk+3d}{2dk-3d}=\frac{d\left(2k+3\right)}{d\left(2k-3\right)}=\frac{2k+3}{2k-3}\left(2\right)\)
Từ (1) , (2) \(\Rightarrow\frac{2a+3b}{2a-3b}=\frac{2c+3d}{2c-3d}\)
b) \(\frac{ab}{cd}=\frac{bk.b}{dk.d}=\frac{b^2}{d^2}\left(1\right)\)
\(\frac{a^2-b^2}{c^2-d^2}=\frac{b^2k^2-b^2}{d^2k^2-d^2}=\frac{b^2\left(k^2-1\right)}{d^2\left(k^2-1\right)}=\frac{b^2}{d^2}\left(2\right)\)
Từ (1) , (2) \(\Rightarrow\frac{ab}{cd}=\frac{a^2-b^2}{c^2-d^2}\)
c) \(\left(\frac{a+b}{c+d}\right)^2=\frac{\left(bk+b\right)^2}{\left(dk+d\right)^2}=\frac{\left[b\left(k+1\right)\right]^2}{\left[d\left(k+1\right)\right]^2}=\frac{b^2.\left(k+1\right)^2}{d^2\left(k+1\right)^2}=\frac{b^2}{d^2}\left(1\right)\)
\(\frac{a^2+b^2}{c^2+d^2}=\frac{b^2k^2+b^2}{d^2k^2+d^2}=\frac{b^2\left(k^2+1\right)}{d^2\left(k^2\right)+1}=\frac{b^2}{d^2}\left(2\right)\)
Từ (1) , (2) \(\Rightarrow\left(\frac{a+b}{c+d}\right)^2=\frac{a^2+b^2}{c^2+d^2}\)
c) có \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a^2}{^{c^2}}=\frac{b^2}{d^2}=\frac{a^2+b^2}{c^2+d^2}\left(1\right)\)
Lại có: \(\frac{a}{c}=\frac{b}{d}=\frac{a+b}{c+d}\Rightarrow\frac{a^2}{c^2}=\frac{b^2}{d^2}=\frac{a^2+b^2}{c^2+d^2}\left(2\right)\)
Từ (1) và (2) có \(\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\frac{a^2+b^2}{c^2+d^2}\left(đpcm\right)\)
các câu còn lại bạn tự làm đi! HI.......
https://bingbe.com/search?category=question&q=Cho%20t%E1%BB%89%20l%E1%BB%87%20th%E1%BB%A9c%20a%20%2Fb%20%3D%20c%20%2Fd%20.%C2%A0Ch%E1%BB%A9ng%20minh%20c%C3%B3%20t%E1%BB%89%20l%E1%BB%87%20th%E1%BB%A9c%20sau%20%3A%0A%0A(%20a%20%2B%20c%C2%A0)2%C2%A0%2F%20(%20b%20%2B%20d%20)2%C2%A0%3D%20a2%C2%A0%20%2B%C2%A0%C2%A0c2%C2%A0%2F%20b2%20%C2%A0%2B%20d%C2%A02%C2%A0%0A%0A(%20Gi%E1%BA%A3%20thi%E1%BA%BFt%20c%C3%A1c%20t%E1%BB%89%20s%E1%BB%91%20%C4%91%E1%BB%81u%20c%C3%B3%20ngh%C4%A9a%20)%C2%A0%0A%0A%C2%A0
Xem ở lick này nhé (mình gửi cho)
Học tốt!!!!!!!!!!!!!
Ta có:
\(\left(\frac{a+b}{c+d}\right)^2\)\(=\frac{\left(a+b\right).\left(a+b\right)}{\left(c+d\right).\left(c+d\right)}\)\(=\frac{a.a+b.b}{c.c+d.d}\)\(=\frac{a^2+b^2}{c^2+d^2}\)
\(\Rightarrow\left(\frac{a+b}{c+d}\right)^2=\frac{a^2+b^2}{c^2+d^2}\).