4.1 Q-18

Question Statement

Find the point that divides the line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ into four equal parts.

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Background and Explanation

To solve this, we will apply the concept of internal division of a line segment. Specifically, we will use the section formula, which allows us to find the coordinates of a point dividing a line segment in a given ratio.

In this case, we need to find points that divide the segment into four equal parts. The three required points divide the segment in the ratios 1:3, 1:1, and 3:1, respectively.

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Solution

We are given two points, $A(x_1, y_1)$ and $B(x_2, y_2)$, and need to find the points that divide the segment joining these points into four equal parts.

1. Point C divides the segment in the ratio 1:3:

Using the section formula:

$$a = \frac{(1)(x_2) + (3)(x_1)}{1 + 3} = \frac{x_2 + 3x_1}{4}$$ $$b = \frac{(1)(y_2) + (3)(y_1)}{1 + 3} = \frac{y_2 + 3y_1}{4}$$

Thus, the coordinates of point C are:

$$C\left[\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right]$$

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2. Point D divides the segment in the ratio 1:1 (midpoint):

The midpoint formula states:

$$D = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Thus, the coordinates of point D are:

$$D\left[\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right]$$

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3. Point F divides the segment in the ratio 3:1:

Again, applying the section formula:

$$e = \frac{(3)(x_2) + (1)(x_1)}{3 + 1} = \frac{3x_2 + x_1}{4}$$ $$f = \frac{(3)(y_2) + (1)(y_1)}{3 + 1} = \frac{3y_2 + y_1}{4}$$

Thus, the coordinates of point F are:

$$F\left[\frac{3x_2 + x_1}{4}, \frac{3y_2 + y_1}{4}\right]$$

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Key Formulas or Methods Used

• Section Formula: To find the point dividing a line segment in a given ratio:

$$\left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)$$

where the point divides the segment in the ratio $m:n$.

• Midpoint Formula: For the midpoint (1:1 ratio):

$$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

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Summary of Steps

1. Point C: Divide in the ratio 1:3 using the section formula:
   • $C\left[\frac{x_2 + 3x_1}{4}, \frac{y_2 + 3y_1}{4}\right]$
2. Point D: Midpoint, divide in the ratio 1:1 using the midpoint formula:
   • $D\left[\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right]$
3. Point F: Divide in the ratio 3:1 using the section formula:
   • $F\left[\frac{3x_2 + x_1}{4}, \frac{3y_2 + y_1}{4}\right]$