To show that a triangle is isosceles, two of the side lengths must be equal. In three dimensions, the distance formula is d = sqrt ((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). In the triangle PQR, the side PQ has the same length as QR. The distance PQ = sqrt ((2-0)^2 +...

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To show that a triangle is isosceles, two of the side lengths must be equal. In three dimensions, the distance formula is d = sqrt ((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). In the triangle PQR, the side PQ has the same length as QR. The distance PQ = sqrt ((2-0)^2 + (6-4)^2 + (5-4)^2) = sqrt (4 + 4 + 1) = sqrt (9) = 3. The distance QR = sqrt ((1-2)^2 + (4-6)^2 + (3-5)^2) = sqrt (1 + 4 + 4) = sqrt (9) = 3. Therefore, PQ = QR and the triangle is shown to be isosceles.

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.**