Question 48568
You can use the distance formula to find the distance between the pairs of given points:
A(1, 1), B(1, 5), C(5, 3)

Distance formula is: {{{d = sqrt((x2-x1)^2+(y2-y1)^2)}}} where the x1 and y1 are the x- and y-coordinates of one of the pair of points and the x2 and y2 are the x- and y-coordinates of the second point.  

Distance between point A(1, 1) = (x1, y1) and B(1, 5) = (x2, y2) is found as follows:
{{{AB = sqrt((1-1)^2+(5-1)^2)}}}
{{{AB = sqrt(4^2)}}}
{{{AB = 4}}}

Distance between point B(1, 5) and C(5, 3) is found by:
{{{BC = sqrt((5-1)^2+(3-5)^2)}}}
{{{BC = sqrt(4^2+(-2)^2)}}}
{{{BC = sqrt(16+4)}}}
{{{BC = sqrt(20)}}}
{{{BC = 2sqrt(5)}}}

The distance between point A(1, 1) and C(5, 3) is found by:
{{{AC = sqrt((5-1)^2+(3-1)^2)}}}
{{{AC = sqrt(4^2+2^2)}}}
{{{AC = sqrt(20)}}}
{{{AC = 2sqrt(5)}}}

As you can see, sides BC and AC are of equal length, therefore the triangle is isosceles.