Test: Section Formula - SAT MCQ

The coordinates of a point dividing the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in the ratio m : n is 
So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) internally in the ratio 2:1 is 

The coordinates of a point dividing the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) externally in the ratio m : n is 
So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) externally in the ratio 2:1 is = (7, 8, 9).

The coordinates of a point dividing the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in the ratio m : n is 
Let ratio be k : 1.
So, z-coordinate of the point will be (k*6 + 1*3)/(k + 1).
We know, for X-Y plane, z coordinate is zero.
(6k + 1 * 3)/(k+1) = 0 ⇒ k = -1/2

Points which trisect the line divides it into 2:1 and 1:2.
The coordinates of a point dividing the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in the ratio m : n is 
For 1:2, coordinates of point are= (7, 15, 4)
For 2:1, coordinates of point are= (10, 21, 0)

We know, midpoint of (x₁, y₁, z₁) and (x₂, y₂, z₂) is ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).
Midpoint of line PR is (5, 4, 6).
Length of median through Q is distance between midpoint of PR and Q i.e.

We know, midpoint of (x₁, y₁, z₁) and (x₂, y₂, z₂) is (x₁+x₂) /2, (y₁+y₂) /2, (z₁+z₂)/2).
So, midpoint of (1, 4, 6) and (5, 8, 10) is ((1+5)/ 2, (4+8)/ 2, (6+10)/2) is (3, 6, 8).

The coordinates of a point dividing the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in the ratio m: n is 
Let the ratio be k : 1.So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) internally in the ratio k: 1 is

⇒ (4k + 1)/(k + 1) = 3
⇒ 4k + 1 = 3k + 3
⇒ k = 2
So, ratio is 2:1.

If coordinates of vertices of a triangle are (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃) the coordinates of centroid of the triangle are ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)
So, coordinates of centroid of the given triangle are ((7 + 5 + 9)/3, (6 + 4 + 5)/3, (4 + 6 + 8)/3) = (7, 5, 3).

Points which trisect the line divides it into 2:1 and 1:2.
The coordinates of a point dividing the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) internally in the ratio m : n is 
For 1:2, coordinates of point are = (7, 15, 4)
For 2:1, coordinates of point are = (10, 21, 0)

We know, midpoint of (x₁, y₁, z₁) and (x₂, y₂, z₂) is ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).
Midpoint of line QR is (5, 5, 4).
Length of median through P is distance between midpoint of QR and P i.e.