English

Prove that the Tetrahedron with Vertices at the Points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a Regular One. - Mathematics

Advertisements
Advertisements

Question

Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.

Solution

The faces of a regular tetrahedron are equilateral triangles.
Let us consider\[\bigtriangleup\]

\[OA = \sqrt{\left( 0 - 0 \right)^2 + \left( 0 - 1 \right)^2 + \left( 0 - 1 \right)^2}\]
\[ = \sqrt{2}\]

\[OB = \sqrt{\left( 1 - 0 \right)^2 + \left( 0 - 0 \right)^2 + \left( 1 - 0 \right)^2}\]
\[ = \sqrt{2}\]
\[AB = \sqrt{\left( 1 - 0 \right)^2 + \left( 0 - 1 \right)^2 + \left( 1 - 1 \right)^2}\]
\[ = \sqrt{2}\]
Hence, this face is an equilateral triangle.
Similarly,\[\bigtriangleup\]OBC,\[\bigtriangleup\]OAC,
\[\bigtriangleup\] ABC all are equilateral triangles.

Hence, the tetrahedron is a regular one.

shaalaa.com
  Is there an error in this question or solution?
Chapter 28: Introduction to three dimensional coordinate geometry - Exercise 28.2 [Page 10]

APPEARS IN

RD Sharma Mathematics [English] Class 11
Chapter 28 Introduction to three dimensional coordinate geometry
Exercise 28.2 | Q 13 | Page 10

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Find the distance between the pairs of points:

(2, 3, 5) and (4, 3, 1)


Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.


Verify the following:

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


Verify the following:

(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.


Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).


Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10.


Find the distance between the following pairs of points: 

P(1, –1, 0) and Q(2, 1, 2)


Using distance formula prove that the following points are collinear:

A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)


Using distance formula prove that the following points are collinear: 

P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)


Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).


Determine the points in yz-plane and are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).


Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle. 


Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.


The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of and are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.


Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1). 


Find the distance of the point whose position vector is `(2hati + hatj - hatk)` from the plane `vecr * (hati - 2hatj + 4hatk)` = 9


Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`


Find the distance of the point (–1, –5, – 10) from the point of intersection of the line `vecr = 2hati - hatj + 2hatk + lambda(3hati + 4hatj + 2hatk)` and the plane `vecr * (hati - hatj + hatk)` = 5.


The distance of a point P(a, b, c) from x-axis is ______.


Find the equation of a plane which is at a distance `3sqrt(3)` units from origin and the normal to which is equally inclined to coordinate axis


Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`


Find the shortest distance between the lines given by `vecr = (8 + 3lambdahati - (9 + 16lambda)hatj + (10 + 7lambda)hatk` and `vecr = 15hati + 29hatj + 5hatk + mu(3hati + 8hatj - 5hatk)`


Find the equation of the plane through the intersection of the planes `vecr * (hati + 3hatj) - 6` = 0 and `vecr * (3hati + hatj + 4hatk)` = 0, whose perpendicular distance from origin is unity.


The distance of the plane `vecr * (2/4 hati + 3/7 hatj - 6/7hatk)` = 1 from the origin is ______.


If one of the diameters of the circle x2 + y2 – 2x – 6y + 6 = 0 is a chord of another circle 'C' whose center is at (2, 1), then its radius is ______.


The points A(5, –1, 1); B(7, –4, 7); C(1, –6, 10) and D(–1, –3, 4) are vertices of a ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×