A square ABCD and PQ are the midpoints of BC and CD. If AP=a and AQ=b .find in terms of a and b the directed line segment AB
The square is presented on the picture:
Denote by "x" the length of the side of the square. "a=|AP|=\\sqrt{|AB|^2+|BP|^2}=\\sqrt{x^2+\\frac{1}{4}x^2}=x\\frac{\\sqrt{5}}{2}". "b=|AQ|=x\\frac{\\sqrt{5}}{2}" . Thus, we received that "a=b=x\\frac{\\sqrt{5}}{2}". Thus, we get: "x=\\frac{2a\\sqrt{5}}{{5}}=\\frac{2b\\sqrt{5}}{{5}}". The length of "AB" is "\\frac{2a\\sqrt{5}}{{5}}=\\frac{2b\\sqrt{5}}{{5}}"
Answer: "\\frac{2a\\sqrt{5}}{{5}}=\\frac{2b\\sqrt{5}}{{5}}"
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