4.1 Q-14

Question Statement

Find the point that is three-fifths of the way along the line segment from $A(-5, 8)$ to $B(5, 3)$ .

Background and Explanation

To solve this problem, we need to find the coordinates of a point that divides the line segment between two points in a specific ratio. The formula used for this is the section formula, which helps determine the coordinates of a point dividing a line segment in a given ratio.

Here, the ratio is 3:2 (since three-fifths of the way means the ratio of distances between the point and the endpoints is 3:2).

Solution

We are given the points $A(-5, 8)$ and $B(5, 3)$ , and we need to find the point that divides the segment in a 3:2 ratio.

Step 1: Use the Section Formula for the $x$ -coordinate

The section formula for a point dividing a line segment in the ratio $m:n$ is given by:

x = \frac{n x_1 + m x_2}{m + n}

For the point dividing the segment in a 3:2 ratio, we substitute the following values:

$x_1 = -5$ (coordinate of point $A$ )
$x_2 = 5$ (coordinate of point $B$ )
$m = 3$ , $n = 2$

Now, we calculate the $x$ -coordinate:

x = \frac{3(5) + 2(-5)}{3 + 2} = \frac{15 - 10}{5} = \frac{5}{5} = 1

So, the $x$ -coordinate of the point is 1.

Step 2: Use the Section Formula for the $y$ -coordinate

Similarly, we use the section formula for the $y$ -coordinate:

y = \frac{n y_1 + m y_2}{m + n}

Substitute the values for the coordinates of $A$ and $B$ :

$y_1 = 8$ (coordinate of point $A$ )
$y_2 = 3$ (coordinate of point $B$ )
$m = 3$ , $n = 2$

Now, we calculate the $y$ -coordinate:

y = \frac{3(3) + 2(8)}{3 + 2} = \frac{9 + 16}{5} = \frac{25}{5} = 5

So, the $y$ -coordinate of the point is 5.

Thus, the point that is three-fifths of the way along the line segment is $C(1, 5)$ .

Key Formulas or Methods Used

Section Formula:
For a point dividing a line segment in the ratio $m:n$ , the coordinates of the point are given by:

x = \frac{n x_1 + m x_2}{m + n}, \quad y = \frac{n y_1 + m y_2}{m + n}

Summary of Steps

Apply the section formula for the $x$ -coordinate using the ratio 3:2.
Apply the section formula for the $y$ -coordinate using the ratio 3:2.
The point dividing the segment is $C(1, 5)$ .