Question

Use Pythagorean theorem to find the missing side c=7, b=6a; find a and b

Answer 1

It might have been helpful to you to have drawn this triangle first, and then labeled the sides.  I will have to assume that the length of the hypotenuse is c = 7, so that a^2 + b^2 = c^2 = 7^2 = 49.

if b = 6a, then this equation becomes a^2 + (6a)^2 = 49.
Simplifying, a^2 + 36a^2 = 49

37a^2 = 49

a^2 = 49/37

a = plus or minus 7 / sqrt(37), or plus or minus 7sqrt(37)/37.

Then b = 6a = 6(7)sqrt(37)/37.  Simplify this to obtain the answer.