Use the geometric mean (leg) theorem. what is the value of a?

Actually Welcome to the concept of Geometric Mean (leg) theorem.

So according to the rule, we know that, here altitude = h and left side is 6 and right side is 14 , so we get as,

==> altitude/left = right/altitude

==> h/6 = 14/h

==> h^2 = 14*6

==> h^2 = 84

2.) so now we apply the Pythagoras theorem in triangle ∆DCB

so we get as,

==> (14)^2 + h^2 = a^2

==> 196 + 84 = a^2

==> 280 = a^2

==> a = √280

==> after simplified => a = √70*4

==> a = 2√70 units

hence the correct answer is a => 2√70 units.

a² + b²= c²

24² + 7² = c²

c= 25