گام به گام کاردرکلاس صفحه 31 درس 2 ریاضی (2) (هندسه)

1)

(الف \(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)

(طرفین وسطین)

\(\begin{array}{l}\frac{a}{b} = \frac{c}{d}\;\;\mathop  \Rightarrow \limits_{bd \ne 0}^{ \times bd} \;\;\frac{a}{b} \times bd = \frac{c}{d} \times bd\\\\ \Rightarrow ad = bc\end{array}\)

(ب \(ad = bc \Rightarrow \frac{a}{b} = \frac{c}{d}\)

(تبدیل حاصل ضرب به تناسب)

\(\begin{array}{l}ad = bc\;\;\mathop  \Rightarrow \limits_{bd \ne 0}^{ \times \frac{1}{{bd}}} \;\;ad \times \frac{1}{{bd}} = bc \times \frac{1}{{dd}}\\\\ \Rightarrow \frac{a}{b} = \frac{c}{d}\end{array}\)

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

(معکوس کردن تناسب)

\(\begin{array}{l}\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\mathop  \Rightarrow \limits_{ac \ne 0}^{ \times \frac{1}{{ac}}} ad \times \frac{1}{{ac}} = bc \times \frac{1}{{ac}}\\\\ \Rightarrow \frac{b}{a} = \frac{d}{c}\end{array}\)

(ت \(\frac{a}{b} = \frac{c}{d} \Rightarrow \left\{ \begin{array}{l}\frac{c}{a} = \frac{d}{b}\\\frac{a}{c} = \frac{b}{d}\end{array} \right.\)

(تعویض جای طرفین یا وسطین)

\(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\left\{ \begin{array}{l}\mathop  \Rightarrow \limits_{ab \ne 0}^{ \times \frac{1}{{ab}}} \;\;\frac{{ad}}{{ab}} = \frac{{bc}}{{ab}} \Rightarrow \frac{c}{a} = \frac{d}{b}\\\mathop  \Rightarrow \limits_{cd \ne 0}^{ \times \frac{1}{{cd}}} \;\;\frac{{ad}}{{cd}} = \frac{{bc}}{{cd}} \Rightarrow \frac{a}{c} = \frac{b}{d}\end{array} \right.\)

(ث \(\frac{a}{b} = \frac{c}{d} \Rightarrow \left\{ \begin{array}{l}\frac{{a + b}}{b} = \frac{{c + d}}{d}\\\frac{a}{{a + b}} = \frac{c}{{c + d}}\end{array} \right.\)

(ترکیب نسبت در صورت یا مخرج)

\(\frac{a}{b} = \frac{c}{d} \Rightarrow \left\{ \begin{array}{l}\frac{a}{b} + 1 = \frac{c}{d} + 1 \Rightarrow \frac{{a + b}}{b} = \frac{{c + d}}{d}\\\frac{b}{a} = \frac{d}{c} \Rightarrow \frac{b}{a} + 1 = \frac{d}{c} + 1 \Rightarrow \frac{{a + b}}{a} = \frac{{c + d}}{c} \Rightarrow \frac{a}{{a + b}} = \frac{c}{{c + d}}\end{array} \right.\)

(ج \(\frac{a}{b} = \frac{c}{d} \Rightarrow \left\{ \begin{array}{l}\frac{{a - b}}{b} = \frac{{c - d}}{d}\\\frac{a}{{b - a}} = \frac{c}{{d - c}}\end{array} \right.\)

(تفضیل نسبت در صورت یا مخرج)

\(\frac{a}{b} = \frac{c}{d} \Rightarrow \left\{ \begin{array}{l}\frac{a}{b} - 1 = \frac{c}{d} - 1 \Rightarrow \frac{{a - b}}{b} = \frac{{c - d}}{d}\\\frac{b}{a} = \frac{d}{c} \Rightarrow \frac{b}{a} - 1 = \frac{d}{c} - 1 \Rightarrow \frac{{b - a}}{a} = \frac{{d - c}}{c} \Rightarrow \frac{a}{{b - a}} = \frac{c}{{d - c}}\end{array} \right.\)

2)

(الف \(\frac{5}{{14}} = \frac{{15}}{{42}} \Rightarrow\) \(5 \times 42 = 15 \times 14\)

(ب  \(3 \times 40 = 12 \times 10 \Rightarrow\) \(\frac{3}{{10}} = \frac{{12}}{{40}}\)

(پ \(\frac{7}{{10}} = \frac{{21}}{{30}} \Rightarrow \frac{{10}}{7} =\) \(\frac{{30}}{{21}}\)

(ت \(\frac{6}{{11}} = \frac{{18}}{{33}} \Rightarrow \frac{6}{{18}} =\) \(\frac{{11}}{{33}}\)  ,  \(\frac{{33}}{{11}} =\) \(\frac{{18}}{6}\)

(ث \(\frac{4}{{14}} = \frac{{10}}{{35}} \Rightarrow \frac{{18}}{{14}} =\) \(\frac{{45}}{{35}}\)  ,  \(\frac{4}{{18}} =\) \(\frac{{10}}{{45}}\)

(ج \(\frac{5}{{12}} = \frac{{10}}{{24}} \Rightarrow \frac{{ - 7}}{{12}} =\) \(\frac{{ - 14}}{{24}}\)  ,  \(\frac{5}{{ - 7}} =\) \(\frac{{10}}{{ - 14}}\)