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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.

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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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34.

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

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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.

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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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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).

Let P (x, y, z) be the required point and
O (0, 0, 0), A (a, 0, 0), B (0, b, 0) and C (0, 0, c) be the given points.
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$open vertical bar PO close vertical bar space equals space open vertical bar PA close vertical bar$
$rightwards double arrow$ $square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space equals space square root of left parenthesis straight x minus straight a right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root$
$rightwards double arrow$ $space space space space straight x squared plus straight y squared plus straight z squared space equals space straight x squared plus straight y squared plus straight z squared minus 2 ax plus straight a squared$
$rightwards double arrow$ $2 ax space equals space straight a squared space rightwards double arrow space straight x space equals space straight a over 2$
 $open vertical bar PO close vertical bar space equals space open vertical bar PB close vertical bar$
$rightwards double arrow$ $square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space equals space square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus straight b right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root$
$rightwards double arrow$ $straight x squared plus straight y squared plus straight z squared space equals space straight x squared plus straight y squared plus straight z squared minus 2 by plus straight b squared space rightwards double arrow space 2 by space equals space straight b squared space rightwards double arrow space straight y space equals space straight b over 2$
Also, $space space space space space space open vertical bar PO close vertical bar space equals space open vertical bar PC close vertical bar$
$rightwards double arrow$ $square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus 0 right parenthesis squared end root space equals space square root of left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared plus left parenthesis straight z minus straight c right parenthesis squared end root$
$rightwards double arrow$ $space space space straight x squared plus straight y squared plus straight z squared space equals space straight x squared plus straight y squared plus straight z squared minus 2 cz plus straight c squared space rightwards double arrow space 2 cz space equals space straight c squared space rightwards double arrow space straight z space equals space straight c over 2$
Hence, the required point is $straight P space left parenthesis straight x comma space straight y comma space straight z right parenthesis space left right arrow space open parentheses straight a over 2 comma space straight b over 2 comma space straight c over 2 close parentheses$

166 Views