Cho \(\displaystyle {a \over b} = {c \over d}\). Chứng minh:
a) \(\displaystyle {{{a^2} - {b^2}} \over {{c^2} - {d^2}}} = {{ab} \over {cd}};\)
b) \(\displaystyle {{{{\left( {a - b} \right)}^2}} \over {{{\left( {c - d} \right)}^2}}} = {{ab} \over {cd}}.\)
Hướng dẫn giải
Áp dụng:
\(\displaystyle {a \over b} = {c \over d} \Rightarrow {a \over c} = {b \over d}\)
\(\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{{a - c}}{{b - d}}\,\left( {b,d,b - d \ne 0} \right)\)
Lời giải chi tiết
a) \(\displaystyle {a \over b} = {c \over d} \Rightarrow {a \over c} = {b \over d}\)
\(\displaystyle \Rightarrow {{ab} \over {cd}}= {a \over c}.{b \over d} = {a \over c}.{a \over c} = {b \over d}.{b \over d} \)
\(\Rightarrow \dfrac{{ab}}{{cd}} = \dfrac{{{a^2}}}{{{c^2}}} = \dfrac{{{b^2}}}{{{d^2}}} \)\(= \dfrac{{{a^2} - {b^2}}}{{{c^2} - {d^2}}}\)
Vậy \(\,\displaystyle {{ab} \over {cd}} = {{{a^2} - {b^2}} \over {{c^2} - {d^2}}}\)
b) \(\displaystyle {a \over b} = {c \over d} \Rightarrow {a \over c} = {b \over d} = {{a - b} \over {c - d}} \)
\(\displaystyle \Rightarrow {{ab} \over {cd}} = {a \over c}.{b \over d} = {{a - b} \over {c - d}}.{{a - b} \over {c - d}} \)\(\,\displaystyle = {{{{\left( {a - b} \right)}^2}} \over {{{\left( {c - d} \right)}^2}}}\)
Vậy \(\displaystyle {{{{\left( {a - b} \right)}^2}} \over {{{\left( {c - d} \right)}^2}}} = {{ab} \over {cd}}.\)
-- Mod Toán 7