Question

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

Answer 1

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: $\frac{2a\sqrt{5}}{{5}}=\frac{2b\sqrt{5}}{{5}}$