01 · TRISECTION
Points of Trisection
Points P and Q are said to trisect line segment AB if they divide AB into three equal parts (AP = PQ = QB).
Point P divides AB in the ratio 1 : 2Point Q divides AB in the ratio 2 : 1
Centroid of a Triangle
The centroid G of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) is:
G = ( (x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3 )
02 · SOLVED NCERT EXAMPLES
Step-by-Step Problem Solving
EXAMPLE 1
Find the coordinates of the points of trisection of line segment joining A(2, -2) and B(-7, 4)
Step 1: Find P (divides AB in ratio 1 : 2):
x = (1(-7) + 2(2)) / (1 + 2) = (-7 + 4) / 3 = -3/3 = -1
y = (1(4) + 2(-2)) / (1 + 2) = (4 - 4) / 3 = 0/3 = 0
P = (-1, 0)
y = (1(4) + 2(-2)) / (1 + 2) = (4 - 4) / 3 = 0/3 = 0
P = (-1, 0)
Step 2: Find Q (divides AB in ratio 2 : 1):
x = (2(-7) + 1(2)) / (2 + 1) = (-14 + 2) / 3 = -12/3 = -4
y = (2(4) + 1(-2)) / (2 + 1) = (8 - 2) / 3 = 6/3 = 2
Q = (-4, 2)
y = (2(4) + 1(-2)) / (2 + 1) = (8 - 2) / 3 = 6/3 = 2
Q = (-4, 2)
The points of trisection are P(-1, 0) and Q(-4, 2).
EXAMPLE 2
If (1, 2), (4, y), (x, 6) and (3, 5) are vertices of a parallelogram taken in order, find x and y
In a parallelogram, diagonals bisect each other (their mid-points are identical!).
Mid-point of AC = Mid-point of BD:
( (1 + x)/2, (2 + 6)/2 ) = ( (4 + 3)/2, (y + 5)/2 )
1 + x = 7 ⟹ x = 6
8 = y + 5 ⟹ y = 3
1 + x = 7 ⟹ x = 6
8 = y + 5 ⟹ y = 3
03 · QUICK SUMMARY
Key Takeaways
01
Trisection points divide the line in ratios 1 : 2 and 2 : 1.
02
Parallelogram property: Mid-point of Diagonal 1 = Mid-point of Diagonal 2.
03
Centroid formula: ( (x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3 ).