Answer to Question #157889 in Geometry for Michael

Question #157889

ABCD is a square and P, Q are the midpoints of BC, CD respectively. If AP = a

and AQ = b, find in terms of a and b, the directed line segments (i) AB


1
Expert's answer
2021-01-26T04:03:00-0500


ABCD is a square: "\\overrightarrow{AB}=\\overrightarrow{DC}, \\overrightarrow{BC}=\\overrightarrow{AD}"

"P, Q" are the midpoints of "BC, CD" respectively: "\\overrightarrow{BP}=\\dfrac{1}{2}\\overrightarrow{BC}, \\overrightarrow{DQ}=\\dfrac{1}{2}\\overrightarrow{DC}"



"\\overrightarrow{AP}=\\overrightarrow{AB}+\\overrightarrow{BP}=\\overrightarrow{AB}+\\dfrac{1}{2}\\overrightarrow{BC}""\\overrightarrow{AQ}=\\overrightarrow{AD}+\\overrightarrow{DQ}=\\overrightarrow{AD}+\\dfrac{1}{2}\\overrightarrow{DC}"

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



"\\overrightarrow{AB}+\\dfrac{1}{2}\\overrightarrow{AD}=\\vec{a}""\\overrightarrow{AD}+\\dfrac{1}{2}\\overrightarrow{AB}=\\vec{b}"


(i)



"\\overrightarrow{AD}=\\vec{b}-\\dfrac{1}{2}\\overrightarrow{AB}""\\overrightarrow{AB}+\\dfrac{1}{2}(\\vec{b}-\\dfrac{1}{2}\\overrightarrow{AB})=\\vec{a}""\\overrightarrow{AB}=\\dfrac{4}{3}\\vec{a}-\\dfrac{2}{3}\\vec{b}"




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