The coordinates for the vertices of a right triangle are (1, 4), (6, 4), and (6,1). What is the length of the longer leg? A) 3 B) 4 C) 5 D) 6

let
A------> (1, 4)
B------> (6, 4)
C----->  (6,1)

we know that
In a right triangle there are  two legs and one hypotenuse

step 1
find the distance AB
d=√[(y2-y1)²+(x2-x1)²]------> d=√[(4-4)²+(6-1)²]-----> d=√25----> 5
AB=5 units

step 2
find the distance BC
d=√[(y2-y1)²+(x2-x1)²]------> d=√[(1-4)²+(6-6)²]-----> d=√9----> 3
BC=3 units

step 3
find the distance AC
d=√[(y2-y1)²+(x2-x1)²]------> d=√[(1-4)²+(6-1)²]-----> d=√34----> 5.83 
AC=5.83 units

therefore
the two legs are
AB=5 units
BC=3 units
the hypotenuse is
AC=5.83 units

the answer is
the length of the longer leg is AB=5 units

see the attached figure