To find the point that divides the line segment joining the points (8, -9) and (2, 3) in the ratio 1:2 internally, we can use the section formula. The section formula states that if a point divides the line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then the coordinates of the point \( (x, y) \) are given by:
\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\]
\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\]
### Step 1: Identify the points and the ratio
Here, we have:
- Point A: \( (x_1, y_1) = (8, -9) \)
- Point B: \( (x_2, y_2) = (2, 3) \)
- Ratio \( m:n = 1:2 \)
### Step 2: Substitute the values into the formula for x-coordinate
Using the formula for the x-coordinate:
\[
x = \frac{1 \cdot 2 + 2 \cdot 8}{1 + 2}
\]
Calculating this step-by-step:
1. Calculate \( 1 \cdot 2 = 2 \)
2. Calculate \( 2 \cdot 8 = 16 \)
3. Add these results: \( 2 + 16 = 18 \)
4. Divide by \( 1 + 2 = 3 \)
So,
\[
x = \frac{18}{3} = 6
\]
### Step 3: Substitute the values into the formula for y-coordinate
Now, using the formula for the y-coordinate:
\[
y = \frac{1 \cdot 3 + 2 \cdot (-9)}{1 + 2}
\]
Calculating this step-by-step:
1. Calculate \( 1 \cdot 3 = 3 \)
2. Calculate \( 2 \cdot (-9) = -18 \)
3. Add these results: \( 3 - 18 = -15 \)
4. Divide by \( 1 + 2 = 3 \)
So,
\[
y = \frac{-15}{3} = -5
\]
### Step 4: Combine the results
The coordinates of the point that divides the line segment in the ratio 1:2 are:
\[
(6, -5)
\]
### Step 5: Determine the quadrant
Now, we need to determine in which quadrant the point \( (6, -5) \) lies.
- The x-coordinate is positive (6).
- The y-coordinate is negative (-5).
This means the point lies in the fourth quadrant.
### Final Answer
The point which divides the line segment joining the points (8, -9) and (2, 3) in the ratio 1:2 internally lies in the fourth quadrant.
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