+91-8076753736[email protected]
logo
logo
-->

Sequences And Series

Question
CBSEENMA11015024
Wired Faculty App

Three vertices of a parallelogram ABCD are A (3, -1, 2), B (1, 2, -4), and C (-1, 1, 2). Find the co-ordinates of the fourth vertex.

Solution

Three vertices of parllelogram ABCD are A (3, -1, 2), B (1, 2, -4), and C (-1, 1, 2).
Let the fourth vertex be D (x, y, z). ABCD is a parallelogram.
rightwards double arrow  Diagonals AC and BD bisect each other rightwards double arrow  Mid-point of AC and BD are coincident.
rightwards double arrow             open parentheses fraction numerator 3 plus left parenthesis negative 1 right parenthesis over denominator 2 end fraction comma space fraction numerator negative 1 plus 1 over denominator 2 end fraction comma space fraction numerator 2 plus 2 over denominator 2 end fraction close parentheses space left right arrow space open parentheses fraction numerator 1 plus straight x over denominator 2 end fraction comma space fraction numerator 2 plus straight y over denominator 2 end fraction comma space fraction numerator negative 4 plus straight z over denominator 2 end fraction close parentheses space
rightwards double arrow                       fraction numerator straight x plus 1 over denominator 2 end fraction space equals space 1 comma space fraction numerator straight y plus 2 over denominator 2 end fraction equals 0 comma space fraction numerator straight z minus 4 over denominator 2 end fraction equals 2
rightwards double arrow                               x = 1,  y = -2,   z = 8
Hence, vertex D is (1, -2, 8)
Using distance formula, we have
                           AC space equals space square root of left parenthesis 3 plus 1 right parenthesis squared plus left parenthesis negative 1 minus 1 right parenthesis squared plus left parenthesis 2 minus 2 right parenthesis squared end root space equals space square root of 16 plus 4 end root space equals space square root of 20 space equals space 2 square root of 5
                      space space space space space space BD space equals space square root of left parenthesis 1 minus 1 right parenthesis squared plus left parenthesis 2 plus 2 right parenthesis squared plus left parenthesis negative 4 minus 8 right parenthesis squared end root space equals space square root of 0 plus 16 plus 144 end root space equals space square root of 160 space equals space 4 square root of 10
                           AC space not equal to space BD
∴ ABCD is not a rectangle.

Some More Questions From Sequences and Series Chapter

Let P (-2, 3, -4) is a point in space. Find the foot of perpendicular from this point to the z-axis.

Let P (-2, 3, -4) is a point in space. Find the the perpendicular distance of P from XY-plane.

Name the octants in which the following points lie:
A (2, 3, 4), B (6, -3, 3), C (2, -1, -6), D (2, 2, -3), E (-1, 3, -6), F (-1, 3, 3), G (-3, -2,5) and H (-1,-2,-5).

The co-ordinates of a point A are (2, -3, 6). Write down the co-ordinates of seven points whose absolute values are the same as those of the co-ordinates of the given point.

Find the octant in which the following points lie:
A (-3,1, 2), B (-3, 1, -2), C (-3, -1,-2), D (3, -1, -2) and E (3,1, 2).

A point lies on the x-axis at a distance of 5 units towards OX'. What are its co-ordinates?

A point lies on z-axis at a distance of 3 units in the direction OZ. Write its co-ordinates.

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation