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Introduction To Three Dimensional Geometry

Short Answer Type

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

119 Views

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

96 Views

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

94 Views

34.

Verify the following:
(0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of a right angled triangle.

113 Views

Long Answer Type

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

126 Views

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.

straight A space left parenthesis 6 comma space 10 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 B space left parenthesis 1 comma space 0 comma space minus 5 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 C space left parenthesis 6 comma space minus 10 comma space 0 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 straight D space left parenthesis 11 comma space 0 comma space 15 right parenthesis space left right arrow space left parenthesis straight x subscript 4 comma space straight y subscript 4 comma space straight z subscript 4 right parenthesis

Using distance formula, we have

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equals space square root of left parenthesis 5 right parenthesis squared plus left parenthesis 10 right parenthesis squared plus left parenthesis 15 right parenthesis squared end root space equals space square root of 25 plus 100 plus 225 end root space equals space square root of 350

...(i)

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space space space space equals space square root of left parenthesis negative 5 right parenthesis squared plus left parenthesis 10 right parenthesis squared plus left parenthesis negative 5 right parenthesis squared end root space equals space square root of 25 plus 100 plus 25 end root space equals square root of 150

...(ii)

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space space space space equals space square root of left parenthesis negative 5 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus left parenthesis negative 15 right parenthesis squared end root space equals space square root of 25 plus 100 plus 225 end root space equals space square root of 350

...(iii)

DA space equals space square root of left parenthesis straight x subscript 4 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 4 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 4 minus straight z subscript 1 right parenthesis squared end root

space space space equals space square root of left parenthesis 5 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus left parenthesis 5 right parenthesis squared end root space equals space square root of 25 plus 100 plus 25 end root space equals space square root of 150

...(iv)

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

equals space square root of left parenthesis 0 right parenthesis squared plus left parenthesis 20 right parenthesis squared plus left parenthesis 10 right parenthesis squared end root space equals space square root of 0 plus 400 plus 100 end root space equals space square root of 500

...(v)

BD space equals space square root of left parenthesis straight x subscript 2 minus straight x subscript 4 right parenthesis squared plus left parenthesis straight y subscript 2 minus straight y subscript 4 right parenthesis squared plus left parenthesis straight z subscript 2 minus straight z subscript 4 right parenthesis squared end root

space space space space space space space space space space equals space square root of left parenthesis negative 10 right parenthesis squared plus left parenthesis 0 right parenthesis squared plus left parenthesis negative 20 right parenthesis squared end root space equals space square root of 100 plus 0 plus 400 end root space equals space square root of 500

...(vi)
From (i), (ii), (iii) and (iv), we have AB = CD, BC = DA
Hence, ABCD is a parallelogram.
Also, from (v) and (vi), AC = BD
∴ AB = CD, BC = DA and AC = BD
Hence, ABCD is a rectangle.

103 Views

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.

100 Views

Short Answer Type

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.

123 Views

Long Answer Type

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.

86 Views

Short Answer Type

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).

166 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

Long Answer Type

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.

Short Answer Type

Long Answer Type

Short Answer Type

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.