Question #157060

ABCD is a square and P, Q are the mid points of BC, CD respectively. If AP = a and AQ = b, find in terms of a and b, the directed line segment AB, AD, BD, and AC.


1
Expert's answer
2021-02-24T07:34:01-0500

ABCDisasquare:AB=DC,BC=ADABCD is a square: \overrightarrow{AB}=\overrightarrow{DC}, \overrightarrow{BC}=\overrightarrow{AD}

P,QarethemidpointsofBC,CDrespectively:P, Qare the midpoints of BC, CD respectively:

BP=12BC,DQ=12DC\overrightarrow{BP} =\frac{1} {2} \overrightarrow{BC}, \overrightarrow{DQ} =\frac{1} {2} \overrightarrow{DC}


AP=AB+BP=AB+12BC\overrightarrow{AP} =\overrightarrow{AB} +\overrightarrow{BP} =\overrightarrow{AB} +\frac{1} {2} \overrightarrow{BC}


AQ=AD+DQ=AD+12DC\overrightarrow{AQ} =\overrightarrow{AD} +\overrightarrow{DQ} =\overrightarrow{AD} +\frac{1} {2} \overrightarrow{DC}



If AP=a\overrightarrow{AP} =\overrightarrow{a} and AQ=b\overrightarrow{AQ} =\overrightarrow{b}



AB+12AD=a\overrightarrow{AB} +\frac{1} {2} \overrightarrow{AD} =\overrightarrow{a}

AD+12AB=b\overrightarrow{AD} +\frac{1} {2} \overrightarrow{AB} =\overrightarrow{b}


i) AD=b12AB\overrightarrow{AD} =\overrightarrow{b} - \frac {1} {2} \overrightarrow{AB}

AB+12b12AB=a\overrightarrow{AB} +\frac{1} {2} \overrightarrow{b} - \frac {1} {2} \overrightarrow{AB}=\overrightarrow{a}


=AB=43a23b=\overrightarrow{AB} =\frac{4}{3} \overrightarrow{a} - \frac {2}{3} \overrightarrow{b}




ii) AD=b12(43a23b)\overrightarrow{AD} =\overrightarrow{b} - \frac {1} {2} (\frac{4}{3} \overrightarrow{a} - \frac {2}{3} \overrightarrow{b})


=AD=43b23a=\overrightarrow{AD} =\frac {4} {3} \overrightarrow{b} - \frac{2}{3} \overrightarrow{a}




iii) ADAB=BD\overrightarrow{AD}-\overrightarrow{AB }=\overrightarrow{BD}


BD=\overrightarrow{BD} = 43b23a\frac {4} {3} \overrightarrow{b} - \frac{2}{3} \overrightarrow{a} - (43a23b)(\frac{4}{3} \overrightarrow{a} - \frac {2}{3} \overrightarrow{b})


BD=2b2a\overrightarrow{BD} =2\overrightarrow{b} -2\overrightarrow{a}



iv) AD+AB=AC\overrightarrow{AD}+\overrightarrow{AB }=\overrightarrow{AC}


AC=\overrightarrow{AC} = 43b23a\frac {4} {3} \overrightarrow{b} - \frac{2}{3} \overrightarrow{a} +(43a23b)+(\frac{4}{3} \overrightarrow{a} - \frac {2}{3} \overrightarrow{b})


AC=23a+23b\overrightarrow{AC} =\frac {2}{3} \overrightarrow{a} + \frac {2}{3} \overrightarrow{b}


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