Test: Dimensional Geometry - 1

Note that if the axes are inclined at 90° , then the distance is given by

Proof: LetP (h,k) be the required point which is equidistant from 0(0,0). A (32,10) and B(42,0).
Then
PO=PA and PO= PB
Now

 

Important Rules
R - 1: If the coordinates of three vertices of a triangle are given then it represents a Right -angled Triangle if the product of the slopes of two adjacent sides is-1.
OR
The square of the largest side equals the sum of the squares of the other two sides.
R - 2: If the coordinates of four vertices of a quadrilateral are known, then it represents a
(i)    Rectangle : If

• opposite sides are equal
• diagonals are equal

(ii)    Square : if

• all sides are equal
• angles are right angles

(iii)    Parallelogram : if any one of the following conditions is true

• opposite sides are equal
• opposite sides are parallel
• middle points of the two diagonals coincide

(iv) Rhombus : if

• adjacent sides arc equal
• adjacent sides are not mutually perpendicular
• opposite sides are parallel Now the solution of the Problem

Proof: Let A(0, -1), B(-2, 3), C(6, 7) and D(8, 3) be the given point. Then using the distance formula, we get

Proof:It is given that the point P(x, y) is equidistant from the points A(3,4)and B(1,-2).
i.e PA-PB

 

Comments about Centroid of a Triangle.

1. It is  the point of intersection of the median of the triangle.
2. It divides each median in the ratio 2:1.
3. It coordinates are

The coordinates of the centroid will be

The three coordinate planes divide the whole space into Eight Parts, called octants. These eight octants and signs of the co-ordinates of a point P in each one of them are shown below : 

Remark : Three fixed lines X'OX Y' O Y and Z' O Z are called x, y and z-axis respectively and their common point of intersection ‘O' is called the origin or pole of coordinates.

We have
1. Distance between two points (x₁, y₁, z₁) and 

2. Distance of a point (x₁, y₁, z₁) from the origin

3. Distance of a point (x₁, y₁, z₁) from x-axis

4. Distance of a point (x₁, y₁, z₁) from y-axis

5. Distance of a point (x₁, y₁, z₁) from z-axis

Note: We have

1. Plane | | to xy piane at a distance c, z = c
2. Plane; | | to t/x plane ai a distance a, x = a
3. Plane | to xx plane at a distance b ,y = b

Remark:- In particular, the equations of YZ, ZX and XY planes are x = 0, y = 0 and z = 0 respectively.

x = a is a plane | | to YZ plane
y = b is a plane | | to ZX plane
Therefore x = a and y = b is a line parallel to z-axis
Note : Remember that two planes intersect in a line

Lengths of the edges will be
p - a, q - b and r - c

The distance between two points in a three dimensional can be calculated as :- 

d = [(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]^½

The coordinates are P(0,7,10), Q(-1,6,6), R(-4,9,6)

 PQ = [(0+1)² + (7-6)² + (10-6)^2]^½

PQ = [1+1+16]^½ 

PQ = [18]^½

QR = [(-1-4)² + (6-9)² + (6-6)^2]^½

QR = [9+9+0]

QR = [18]^½

PR = [(0+4)² + (7-9)² + (10-6)^2]^½

PR = [16 + 4 + 16]^½

PR = 6

As we can see that (PR)² = (QR)² + (PQ)² 

I.e. isoceles and right triangle

Centre o f the sphere = ( 1 , 3 , 4 )
Sphere passes through (3. 2, 2)
∴ Radius

Remark: Sphere passes through four points.
Note: Distance of any one of the ratio m : n arc given as follows

The coordinates (x, y, z) of a point P which divides the line joining A(x₁, y₁ z₁ ) and B(x₂, y₂, z₂) in the ratio m : n are given as follows:
1. P divides AB internally in the ratio m : n
Then

2. P divides AB externally in the ratio m : n

Now yz plane is given by x = 0. It means that suppose the point whose x-coordinate is zero divides the line joining (2, 4, 5) and (3. 5, -6) in the ratio m : n.
Then

Centroid of the triangle whose vertices are (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃) are given by 
In this question, the vertices of the triangle are (3, 2, 0), (5, 3. 2) and (-9, 6, -3).
∴The coordinates of the centroid are

About Tetrahedron
1. Tetrahedron is a pyramid with a triangular base.
2. It has four vertices.
3. It has four faces. If BCD is the triangular base and A the top ertex, then the four faces are BCD, ABC, ACDand ABD
4. It has six edges, AB, AC, AD, BC, CD and DB are its six edges.
5. If (x₁,y₁,z₁) , (x₂,y₂,z₂), (x₃,y₃,z₃) and (x₄,y₄,z₄) are the coordinates of the vertices, then the Coordinates of its centroid (centre of gravity) are


above figure shows a tetrahedron. Rase triangle is BCD. The centre of gravity G of the tetrahedron divides the line AG. in the ratio 3:1, where G, is the centre of gravity of the base triangle BC (E is the middle point of DC and Gt divides BE in :he ratio 2 : 1).
6. The volume V of the tetrahedron is given by