Show that P(0,4,4), Q(2,6,5) and R(1,4,3) are vertices of an isosceles triangle.

An isosceles triangle has two equal sides. The distance
between the points X(x1, y1, z1) and Y(x2, y2, z2) is given by XY = sqrt[(x1 - x2)^2 +
(y1 - y2)^2 + (z1 - z2)^2]

Here the vertices of the
triangle are P(0,4,4), Q(2,6,5) and R(1,4,3)

PQ =
sqrt[(0-2)^2 + (4 - 6)^2 + (4 - 5)^2]

= sqrt[4 + 4 +
1]

= sqrt 9

=
3

QR = sqrt[(2 - 1)^2 + (6 - 4)^2 + (5 -
3)^2]

= sqrt[1+ 4 + 4]

= sqrt
9

= 3

RP = sqrt[(1 - 0)^2 + (4
- 4)^2 + (3 - 4)^2]

= sqrt[1 + 0 +
1]

= sqrt 2

The length of two
sides PQ and QR is equal.

Therefore the
triangle PQR is isosceles.