What expression represents the value of v? v equals the square root of quantity z times y end quantity v equals the square root of quantity w times z end quantity v equals the square root of quantity w times the sum of w plus z end quantity v equals the square root of quantity z times the sum of w plus z end quantity

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.

option: d is correct

step-by-step explanation: