Answer: Fourth option:
v = √( z*(z + w ) )
Step-by-step explanation:
Ok, v is the hypotenuse of the larger triangle, we know, by the Pythagorean's theorem, that:
v = √(y^2 + z^2)
now, in the smaller triangle we have that:
x = √(y^2 + w^2)
And we also know that:
z + w = √(x^2 + v^2)
(z + w)^2 = x^2 + v^2
and x^2 = (y^2 + w^2)
(z + w)^2 = z^2 + w^2 + 2*z*w = y^2 + w^2 + v^2
v = √( z^2 + w^2 + 2*z*w - y^2 - w^2)
v = √( z^2 + 2*z*w - y^2 )
and y is the altitude of the larger triangle rectangle, so, by the geometric mean theorem, we have that:
y^2 = z*w
v = √( z^2 + 2*z*w - z*w ) = √( z^2 + z*w ) = √( z*(z + w ) )
The correct option is the fourth one.
