Questions: Find the coordinates of P if it is 1/5 of the distance from A(-3,-2) to B(12,3)

Transcript text: Find the coordinates of $P$ if it is $\frac{1}{5}$ of the distance from $A(-3,-2)$ to $B(12,3)$

Solution

Solution Steps

To find the coordinates of point $ P $ that is $\frac{1}{5}$ of the distance from $ A(-3, -2) $ to $ B(12, 3) $, we can use the section formula. The section formula helps in finding a point that divides a line segment in a given ratio. Here, the ratio is $ \frac{1}{5} $ of the total distance, which means $ P $ divides the line segment $ AB $ in the ratio $ 1:4 $.

Step 1: Determine the Coordinates of Points A and B

The coordinates of point $ A $ are given as $ A(-3, -2) $ and the coordinates of point $ B $ are $ B(12, 3) $.

Step 2: Identify the Ratio

Point $ P $ divides the line segment $ AB $ in the ratio $ 1:4 $. This means that $ P $ is $\frac{1}{5}$ of the way from $ A $ to $ B $.

Step 3: Apply the Section Formula

Using the section formula, the coordinates of point $ P $ can be calculated as follows: \[ P_x = \frac{m \cdot B_x + n \cdot A_x}{m + n} = \frac{1 \cdot 12 + 4 \cdot (-3)}{1 + 4} = \frac{12 - 12}{5} = 0 \] \[ P_y = \frac{m \cdot B_y + n \cdot A_y}{m + n} = \frac{1 \cdot 3 + 4 \cdot (-2)}{1 + 4} = \frac{3 - 8}{5} = \frac{-5}{5} = -1 \]

Step 4: Final Coordinates of Point P

Thus, the coordinates of point $ P $ are $ P(0, -1) $.