To find the coordinates of point $P$ that divides the line segment $AB$ in a ratio of $3:2$, we can use the section formula or think about it as moving a certain fraction of the way from $A$ to $B$.
Step 1: Identify the coordinates and the ratio.
* Point $A = (3, 4)$
* Point $B = (6, 10)$
* Ratio $AP : PB = 3 : 2$
This means the total number of "parts" in the segment is $3 + 2 = 5$.
Point $P$ is located $\frac{3}{5}$ of the way from $A$ to $B$.
Step 2: Calculate the change in x and y.
First, let's find the total distance (change) in the x-coordinates and y-coordinates from $A$ to $B$.
*
Change in x: $x_B - x_A = 6 - 3 = 3$
*
Change in y: $y_B - y_A = 10 - 4 = 6$
Step 3: Calculate the coordinates of P.
We need to add $\frac{3}{5}$ of these changes to the starting coordinates of $A$.
For the x-coordinate:
$$x_P = x_A + \frac{3}{5}(\text{change in } x)$$
$$x_P = 3 + \frac{3}{5}(3)$$
$$x_P = 3 + \frac{9}{5}$$
Convert $3$ to a fraction with a denominator of 5: $3 = \frac{15}{5}$.
$$x_P = \frac{15}{5} + \frac{9}{5} = \frac{24}{5}$$
$$x_P = 4.8$$
For the y-coordinate:
$$y_P = y_A + \frac{3}{5}(\text{change in } y)$$
$$y_P = 4 + \frac{3}{5}(6)$$
$$y_P = 4 + \frac{18}{5}$$
Convert $4$ to a fraction with a denominator of 5: $4 = \frac{20}{5}$.
$$y_P = \frac{20}{5} + \frac{18}{5} = \frac{38}{5}$$
$$y_P = 7.6$$
Verification:
Let's check if the distances match the ratio.
Distance $A$ to $P$ in x: $4.8 - 3 = 1.8$
Distance $P$ to $B$ in x: $6 - 4.8 = 1.2$
Ratio $1.8 : 1.2$. Divide both by $0.6$: $3 : 2$. This matches.
Distance $A$ to $P$ in y: $7.6 - 4 = 3.6$
Distance $P$ to $B$ in y: $10 - 7.6 = 2.4$
Ratio $3.6 : 2.4$. Divide both by $1.2$: $3 : 2$. This matches.
The calculations are correct.
Final Answer:
(4.8, 7.6)
Parent Tip: Review the logic above to help your child master the concept of partitioning line segments in two dimensions practice worksheet.
