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31. Verify the following : (0, 7, -10), (1, 6, -6) and ( 4, 9, -6) are the vertices of an isosceles triangle.

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32. Show that A (a, b, c), B (b, c, a) and C (c, a, b) are vertices of an equilateral triangle.

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33. Show that A (6, 10, 10), B (1, 0, -5) and C (6, -10, 0) are the vertices of a right angled triangle.

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straight P space left parenthesis 0 comma space 7 comma space 10 right parenthesis space left right arrow space left parenthesis straight x subscript 1 comma space straight y subscript 1 comma space straight z subscript 1 right parenthesis straight Q space left parenthesis negative 1 comma space 6 comma space 6 right parenthesis space left right arrow space left parenthesis straight x subscript 2 comma space straight y subscript 2 comma space straight z subscript 2 right parenthesis straight R space left parenthesis negative 4 comma space 9 comma space 6 right parenthesis space left right arrow space left parenthesis straight x subscript 3 comma space straight y subscript 3 comma space straight z subscript 3 right parenthesis

Using distance formula, we have

space space space space space space space space space space space PQ equals square root of left parenthesis straight x subscript 1 minus straight x subscript 2 right parenthesis squared plus left parenthesis straight y subscript 1 minus straight y subscript 2 right parenthesis squared plus left parenthesis straight z subscript 1 minus straight z subscript 2 right parenthesis squared end root

space space space equals square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space equals space square root of 18 space equals space 3 square root of 2

...(i)

QR space equals space square root of left parenthesis straight x subscript 2 minus straight x subscript 3 right parenthesis squared plus left parenthesis straight y subscript 2 minus straight y subscript 3 right parenthesis squared plus left parenthesis straight z subscript 2 minus straight z subscript 3 right parenthesis squared end root

equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis negative 3 right parenthesis squared plus left parenthesis 0 right parenthesis squared end root space equals space square root of 18 space equals space 3 square root of 2

...(ii)

space space space CA equals square root of left parenthesis straight x subscript 3 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 3 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 3 minus straight z subscript 1 right parenthesis squared end root

space space space equals square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root space equals space square root of 36 space equals space 6

...(iii)
From (i), (ii) and (iii), we have
PQ = QR
and

PQ squared plus QR squared space equals space 18 plus 18 equals 36 space equals space RP squared

Hence, triangle PQR is an right angled triangle.

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35. Verify the following:
(-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram.

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36. Verify the following:
Show that A (6, 10, 10), B (1, 0, -5), C (6, -10, 0) and D (11, 0, 15) are the vertices of a rectangle.

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37. Show that points A (1, 2, 3), B (-1, -2, -1), C (2, 3, 2) and D (4, 7, 6) are the vertices of a parallelogram ABCD, but it is not a rectangle.

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38.

Show that the points:
P (-1,-6, 10), Q (1, -3, 4), R (-5, -1, 1) and S (-7, -4, 7) are the vertices of a rhombus.

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39. Show that the points:
A (-2, 6, -2), B (0, 4, -1), C (-2, 3, 1) and D (-4, 5, 0) are the vertices of a square.

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40. Find the co-ordinates of a point equidistant from points A (a, 0, 0), B (0, b, 0), C (0, 0, c) and O (0, 0, 0).

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