Three Dimensional Geometry Test

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Three Dimensional Geometry Test - 2

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Question 1
2 / -0.83

The coordinates of the centroid of a triangle whose vertices are (2, –1, 6), (3, 3, –2) and (–2, 1, –1) are:

A
(1, 1, 1)

B

C

D

Solution

Centroid of a triangle with vertices (x₁ ,y₁ );(x₂ ,y₂ ) and (x₃ ,y₃ ) is calculated by the formula (x₁ +x₂ +x₃ )/3 , (y₁ +y₂ +y₃ )/3, (z₁ +z₂ +z₃ )/3
So, centroid =[(2+3-2)/3 , (-1+3+1)/3 , (6-2-1)/3]
= [1,1,1]
Question 2
2 / -0.83

The ratio, in which YZ-plane divides the line joining (2, 4, 5) and (3, 5, 7) is

A
5 : 4

B
2 : 3

C
1 : 2

D
4:1

Solution

Let the YZ plane divide the line segment joining points (2,4,5) and (3,5,7) in the ratio k:1.
Hence, by section formula, the coordinates of point of intersection are given by :
[k(3)+2]/(k+1), [k(5)+4]/(k+1), [k(8)+7]/(k+1)
On the YZ plane, the x-coordinate of any point is zero.
(3k+2)/(k+1) = 0
3k + 2 = 0
k = -2/3
Question 3
2 / -0.83

The ratio in which the join of points (1, –2, 3) and (4, 2, –1) is divided by XOY plane is:

A
- 1 : 3

B
- 3 : 1

C
3 : 1

D
1 : 3

Solution
Question 4
2 / -0.83

Three vertices of a parallelogram PQRS are P(3, –1, 2), Q (1, 2, –4) and R (- 1, 1, 2). Find the coordinates of the fourth vertex.

A
(1,-2,-8)

B
(1,-2,8)

C
(1,2,8)

D
(-1,-2,8)

Solution
Question 5
2 / -0.83

A point R with x-coordinate 1 lies on the line segment joining the points P(-2, 3,5) and Q (7, 0, -1). The coordinates of the point R are

A
(1,-2,3)

B
(-1,2,3)

C
(-1,-2,-3)

D
(1,2,3)

Solution

The coordinates of points P and Q are given as P(2,−3,5) and (7,0,-1)
Let R divide line segment PQ in the ratio k:1
Hence by section formula, the coordinates of point R are given by,
(k(7)+2/k+1,k(0)−3/k+1, k(-1)+5/k+1)
=(7k+2/k+1, −3/k+1, -1k+5/k+1)
It is given that the x-coordinate of point R is 1.
∴7k+2/k+1=1
⇒7k+2=k+1
⇒6k=-1
⇒k=-1/6
Hence the coordinates of R are (1,2,3).
Question 6
2 / -0.83

If the origin is the centroid of the triangle ABC with vertices A (2a, 14, 6), B (8, 3b, -10) and C(-4, 2, 2c), then the values of a and c are.

A
2 and -2

B
-2 and 2

C
1/2 and 1/2

D

Solution

The coordinates of the centroid of △ABC
=[(2a −8+4)/3 , (3b+14+2)0/3 , (6 −10+2c)/3]
=[(2a-4)/3 , (3b+16)/3 , (2c −4)/3]
It is given that origin is the centroid of △ABC
∴(0,0,0)=[(2a+4)/3 , (3b+16)/3 , (2c −4)/3]
(2a+4)/3 = 0 , (3b+16)/3 = 0and (2c −4)/3 = 0
⇒a=−2 and c=2
Question 7
2 / -0.83

If the origin is the centroid of the triangle PQR with vertices P(2a, 2, 6), Q(-4, 3b, -10) and (8, 14, 2c), then the values of a, b and c are:

A

B

C

D

Solution

The coordinates of the centroid of △PQR
=[(2a −4+8)/3 , (2+3b+14)/3 , (6 −10+2c)/3] =((2a+4)/3, (3b+16)/3 ,(2c −4)/3)
It is given that origin is the centroid of △PQR
∴(0,0,0)=((2a+4)/3, (3b+16)/3, (2c −4)/3)
⇒(2a+4)/3 =0, (3b+16)/3 = 0 and (2c −4)/3=0
⇒a =−2,b =−16/3 and c = 2
Question 8
2 / -0.83

The ratio in which the join of points (1, –2, 3) and (4, 2, –1) is divided by XOY plane is:

A
3 : 1

B
1 : 3

C
–3 : 1

D
–1 : 3

Solution

Let P be the point where the line joining the given two points (1,−2,3) and (4,2,−1) intersects the X −Y plane in m:n ratio. We are to find m:n.
Now the co-ordinate of the point P be [(4m+n)/m+n , (2m −2n)/m+n , (−m+3n)/m+n)].
As the point P lies on the X −Y plane, (−m+4n)/m+n = 0
or, −m+3n=0
or, m/n = 3/1
or, m:n = 3:1
Question 9
2 / -0.83

The coordinates of the point R which divides the line segment joining two points P (x₁ , y₁ , z₁ ) and Q (x₂ , y₂ , z₂ ) externally in the ratio m : n are given by

A

B

C

D

Solution

Using section formula
The coordinates of point R that divides the line segment joining points P (x₁ , y₁ , z₁ ) and Q (x₂ , y₂ , z₂ ) externally in the ratio m: n are .
Question 10
2 / -0.83

The ratio in which the line joining the points (1, 2, 3) and (-3.4, -5) is divided by the xy-plane is:

A
3 : 5

B
2 : 5

C
5 : 2

D
5 : 3

Solution

Since line is divided by xy plane
so z=0
for z axis
0=-5k+3 (1)/k+1
-5k+3=0
k=3/5
ratio = 3:5