Answer to Question #157018 in Analytic Geometry for Stephen

Question

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

Answer 1

We have that

$a=AB+BC/2.......(1)\\ b=AD+DC/2.......(2)$

But, AD=BC and AB=DC

So,

$a=DC+BC/2\\ b=BC+DC/2$

Solving the simultaneous equation, we have that:

$DC=\frac{2}{3}(2a-b), BC=\frac{2}{3}(2b-a)$

Also,

$AB=DC=\frac{2}{3}(2a-b)\\ AD=BC=\frac{2}{3}(2b-a)$

$AC=AB+BC=\frac{2}{3}(a+b)\\ BD=AD-AB=2(b-a)$