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

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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.

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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.

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