Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let A(3,3,3) , B(0,6,3) , C( 1,7,7) and D (4,4,7) are the vertices of quadrilateral \[\square ABCD\]
We have :
AB =\[\sqrt{\left( 0 - 3 \right)^2 + \left( 6 - 3 \right)^2 + \left( 3 - 3 \right)^2}\]
\[ = \sqrt{9 + 9 + 0}\]
\[ = \sqrt{18}\]
\[ = 3\sqrt{2}\]
BC =\[\sqrt{\left( 1 - 0 \right)^2 + \left( 7 - 6 \right)^2 + \left( 7 - 3 \right)^2}\]
\[= \sqrt{\left( 1 \right)^2 + \left( 1 \right)^2 + \left( 4 \right)^2}\]
\[ = \sqrt{1 + 1 + 16}\]
\[ = \sqrt{18}\]
\[ = 3\sqrt{2}\]
CD =\[\sqrt{\left( 4 - 1 \right)^2 + \left( 4 - 7 \right)^2 + \left( 7 - 7 \right)^2}\]
\[= \sqrt{\left( 3 \right)^2 + \left( - 3 \right)^2 + \left( 0 \right)^2}\]
\[ = \sqrt{9 + 9 + 0}\]
\[ = \sqrt{18}\]
\[ = 3\sqrt{2}\]
DA =\[\sqrt{\left( 4 - 3 \right)^2 + \left( 4 - 3 \right)^2 + \left( 7 - 3 \right)^2}\]
\[ = \sqrt{1 + 1 + 16}\]
\[ = \sqrt{18}\]
\[ = 3\sqrt{2}\]
AB = BC = CD = DA
AC =
AB = BC = CD = DA
AC =\[\sqrt{\left( 1 - 3 \right)^2 + \left( 7 - 3 \right)^2 + \left( 7 - 3 \right)^2}\]
\[ = \sqrt{4 + 16 + 16}\]
\[ = \sqrt{36}\]
\[ = 6\]
\[\]
\[ = \sqrt{16 + 4 + 16}\]
\[ = \sqrt{36}\]
\[ = 6\]
Therefore, the points are the vertices of a square.
APPEARS IN
संबंधित प्रश्न
If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c.
Name the octants in which the following points lie:
(4, –3, 5)
Name the octants in which the following points lie:
(7, 4, –3)
Find the image of:
(–5, 4, –3) in the xz-plane.
Planes are drawn through the points (5, 0, 2) and (3, –2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.
Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
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.
Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
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).
If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.
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 an isosceles triangle.
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.
The coordinates of the mid-points of sides AB, BC and CA of △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.
Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)
A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/γ` = 3
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
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 ______.
Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.
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
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
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
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 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 ______.
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.
