Question

Find the coordinates of the points of trisection P and Q of the line-segment AB as shown in the given figure.

Answer 1

Trisection means dividing a line segment into three equal parts.

∴ PA = PQ = QB.

Let point P divides line segment AB in ratio 1 : 2.

Let (x₁, y₁) = (–1, 1), (x₂, y₂) = (4, 8) and m₁ = 1, m₂ = 2

∴ By section formula,

Point P = (P_x, P_y)

= `((m_1x_2 + m_2x_1)/(m_1 + m_2) + (m_1y_2 + m_2y_1)/(m_1 + m_2))`

= `((1(4) + 2(1))/(1 + 2), (1(8) + 2(1))/(1 + 2))`

= `((4 - 2)/3, (8 + 2)/3)`

`(P_x, P_y) = (2/3, 10/3)`

Let point Q divides line segment AB in the ratio 2 : 1

Let (x₁, y₁) = (–1, 1), (x₂, y₂) = (4, 8) and m₁ = 2, m₂ = 1

∴ By section formula,

Point Q = (Q_x, Q_y)

= `((m_1x_2 + m_2x_1)/(m_1 + m_2) + (m_1y_2 + m_2y_1)/(m_1 + m_2))`

= `((2(4) + 1(1))/(2 + 1), (2(8) + 1(1))/(2 + 1))`

= `((8 - 1)/3, (16 + 1)/3)`

`(Q_x, Q_y) = (7/3, 17/3)`