Let ABCD be a square, and EFGH be a square whose area is twice that of ABCD. Prove that the ratio of AB to AC is equal to the ratio of AB to EF

Answer 1

if the second square is twice the area, then the sides for the large square must be sqrt2 times as large.

EF=AB*sqrt2
EF/AB =sqrt2
Now consider the smaller square. ABC forms a right triangle, sides AB, CD, and AC
but CD=AB so AC by Pythagorean theorem is AC=sqrt(AB^2+AB^2)=sqrt(2AB^2
=AB*sqrt2
AC/AB = sqrt2
so the Ration of AB to AC is the same as ratio of AB to AC

Answer 2

let the side of square ABCD be x
then area = x^2

area of square EFGH = 2x^2
side of square EFGH = √2 x
EF = √2 x

AC^2 = x^2 + x^2 = 2x^2
AC = √2 x

AB/AC = x/√2x = 1/√2
AB/EF = x/√2x = 1/√2

thus AB/AC = AB/EF