Questions: Given points A(-2,4) and B(7,-2), find the coordinates of the point P on the directed line of AB that partitions AB in the ratio of 1:2.

Transcript text: 2) Given points $A(-2,4)$ and $B(7,-2)$, find the coordinates of the point $P$ on the directed line of $A B$ that partitions $A B$ in the ratio of $1: 2$.

Solution

Solution Steps

To find the coordinates of the point $ P $ that partitions the line segment $ AB $ in the ratio $ 1:2 $, we can use the section formula. The section formula for a point dividing a line segment in the ratio $ m:n $ is given by: \[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] Here, $ A(x_1, y_1) = (-2, 4) $ and $ B(x_2, y_2) = (7, -2) $, and the ratio $ m:n = 1:2 $.

Step 1: Identify Points and Ratio

We are given points $ A(-2, 4) $ and $ B(7, -2) $ and need to find the coordinates of point $ P $ that divides the line segment $ AB $ in the ratio $ 1:2 $.

Step 2: Apply the Section Formula

Using the section formula, the coordinates of point $ P $ that divides the segment $ AB $ in the ratio $ m:n $ are given by: \[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] Substituting $ m = 1 $, $ n = 2 $, $ x_1 = -2 $, $ y_1 = 4 $, $ x_2 = 7 $, and $ y_2 = -2 $: \[ P(x, y) = \left( \frac{1 \cdot 7 + 2 \cdot (-2)}{1+2}, \frac{1 \cdot (-2) + 2 \cdot 4}{1+2} \right) \]

Step 3: Calculate Coordinates

Calculating the $ x $-coordinate: \[ P_x = \frac{7 - 4}{3} = \frac{3}{3} = 1.0 \] Calculating the $ y $-coordinate: \[ P_y = \frac{-2 + 8}{3} = \frac{6}{3} = 2.0 \]