To find the coordinates of point $P$ that divides the line segment from $A(9, -8)$ to $B(-6, 7)$ in the ratio 2:1 (since point $P$ is two-thirds the length of the segment from $A$ to $B$), we can use the section formula. The section formula is:

$x = \frac{m}{m+n}(x_2 - x_1) + x_1$ $y = \frac{m}{m+n}(y_2 - y_1) + y_1$

Where:

• $A(x_1, y_1) = (9, -8)$
• $B(x_2, y_2) = (-6, 7)$
• The ratio $m:n$ is 2:1 (since point $P$ divides the segment in a 2:1 ratio)

Step 1: Find the x-coordinate

Substitute the values into the formula:

$x = \frac{2}{2+1}(-6 - 9) + 9$ $x = \frac{2}{3}(-15) + 9$ $x = -10 + 9 = -1$

Step 2: Find the y-coordinate

Substitute the values into the formula for the y-coordinate:

$y = \frac{2}{2+1}(7 - (-8)) + (-8)$ $y = \frac{2}{3}(7 + 8) - 8$ $y = \frac{2}{3}(15) - 8$ $y = 10 - 8 = 2$

Thus, the coordinates of point $P$ are $(-1, 2)$.

Final Answer:

The coordinates of point $P$ are $(-1, 2)$, so the correct option is:

(–1, 2).

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Would you like more details or clarification?

Here are 5 related questions:

1. How do you use the section formula for different ratios (other than 2:1)?
2. How do you calculate the length of a line segment between two points on the coordinate plane?
3. What is the significance of the section formula in geometry?
4. How do you find the midpoint of a line segment?
5. What happens to the coordinates of point P if the ratio is reversed (1:2 instead of 2:1)?

Tip: The section formula can be used to find any point on a line segment divided in a given ratio, whether internal or external.