Advertisements
Advertisements
Question
Show that the 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 not a rectangle.
Advertisements
Solution
To show that ABCD is a parallelogram, we need to show that its two opposite sides are equal.
\[AB = \sqrt{\left( - 1 - 1 \right)^2 + \left( - 2 - 2 \right)^2 + \left( - 1 - 3 \right)^2}\]
\[ = \sqrt{4 + 16 + 16}\]
\[ = \sqrt{36}\]
\[ = 6\]
\[BC = \sqrt{\left( 2 + 1 \right)^2 + \left( 3 + 2 \right)^2 + \left( 2 + 1 \right)^2}\]
\[ = \sqrt{9 + 25 + 9}\]
\[ = \sqrt{43}\]
\[CD = \sqrt{\left( 4 - 2 \right)^2 + \left( 7 - 3 \right)^2 + \left( 6 - 2 \right)^2}\]
\[ = \sqrt{4 + 16 + 16}\]
\[ = \sqrt{36}\]
\[ = 6\]
\[DA = \sqrt{\left( 1 - 4 \right)^2 + \left( 2 - 7 \right)^2 + \left( 3 - 6 \right)^2}\]
\[ = \sqrt{9 + 25 + 9}\]
\[ = \sqrt{43}\]
\[AB = CD and BC = DA\]
\[\text{ Since, opposite pairs of sides are equal } . \]
\[ \therefore \text{ ABCD is a parallelogram }\]
\[AC = \sqrt{\left( 2 - 1 \right)^2 + \left( 3 - 2 \right)^2 + \left( 2 - 3 \right)^2}\]
\[ = \sqrt{1 + 1 + 1}\]
\[ = \sqrt{3}\]
\[BD = \sqrt{\left( 4 + 1 \right)^2 + \left( 7 + 2 \right)^2 + \left( 6 + 1 \right)^2}\]
\[ = \sqrt{25 + 81 + 49}\]
\[ = \sqrt{155}\]
\[\text{ Since }, AC \neq BD\]
Thus, ABCD is not a rectangle.
APPEARS IN
RELATED QUESTIONS
Name the octants in which the following points lie:
(–5, –3, –2)
Find the image of:
(–2, 3, 4) in the yz-plane.
Find the image of:
(5, 2, –7) in the xy-plane.
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.
Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Write the distance of the point P (2, 3,5) from the xy-plane.
Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.
Find the ratio in which the line segment joining the points (2, 4,5) and (3, −5, 4) is divided by the yz-plane.
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
Find the co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).
If a line makes an angle of `pi/4` with each of y and z axis, then the angle which it makes with x-axis is ______.
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.
Find the equations of the line passing through the point (3,0,1) and parallel to the planes x + 2y = 0 and 3y – z = 0.
Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.
The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is ax + by `+- (sqrt(a^2 + b^2) tan alpha)z ` = 0
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
If l1, m1, n1 ; l2, m2, n2 ; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
The locus represented by xy + yz = 0 is ______.
The direction cosines of the vector `(2hati + 2hatj - hatk)` are ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vecr.(5hati - 3hatj - 2hatk)` = 38.
