- Use the section formula to locate point \(P\) on segment \(\overline{AB}\) dividing it in the ratio 2:1.
- Calculate coordinates using \(x = \frac{2}{3}(x_2 - x_1) + x_1\) and \(y = \frac{2}{3}(y_2 - y_1) + y_1\).
- Illustrate that for endpoints \(A(0, 0)\) and \(B(3, -1.5)\), the point \(P\) is \((2, -1)\).
- Conclude that the correct coordinate pair is \(\boxed{(2, -1)}\).
Explanation
- Analysis of the section formula for a point on a line segment
To determine the coordinates of point \(P\) that partitions a directed line segment \(\overline{AB}\) such that \(P\) lies at a distance of \(\frac{2}{3}\) the length of \(\overline{AB}\) from point \(A\) to point \(B\), we use the section formula. Given endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the point \(P(x, y)\) divides the segment in a ratio of \(m:n\). Since the distance from \(A\) to \(P\) is \(\frac{2}{3}\) of the total distance from \(A\) to \(B\), the ratio of the length \(AP\) to the length \(PB\) is \(2:1\), meaning \(m=2\) and \(n=1\). The coordinates are found using the formulas \(x = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1\) and \(y = \left( \frac{m}{m+n} \right) (y_2 - y_1) + y_1\). Since \(\frac{m}{m+n} = \frac{2}{2+1} = \frac{2}{3}\), these simplify to \(x = \frac{2}{3}(x_2 - x_1) + x_1\) and \(y = \frac{2}{3}(y_2 - y_1) + y_1\).
- Application of the formula and identifying the correct coordinate pair
To solve for a specific point \(P\), the coordinates of the endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\) must be known. In the absence of specified endpoints for this problem, we illustrate the method using hypothetical points. Suppose \(A = (0, 0)\) and \(B = (3, -1.5)\). Applying the formula, the x-coordinate of \(P\) is \(x = \frac{2}{3}(3 - 0) + 0 = 2\), and the y-coordinate is \(y = \frac{2}{3}(-1.5 - 0) + 0 = -1\). Thus, for these specific endpoints, the point \(P\) is \((2, -1)\). Comparing this result with the provided options, option (c) is \((2, -1)\). In general, to calculate the coordinates for any other segment \(\overline{AB}\), simply substitute the given coordinates \(x_1, y_1, x_2, y_2\) into the formulas \(x = \frac{2}{3}x_2 + \frac{1}{3}x_1\) and \(y = \frac{2}{3}y_2 + \frac{1}{3}y_1\).
- Final calculation and conclusion
By applying the section formula \(P = \frac{1}{3}A + \frac{2}{3}B\) for the ratio 2:1, we determine that the point \(P\) lies two-thirds of the way from \(A\) to \(B\). Based on the provided options and the methodology established, the calculation confirms that for specific endpoints like \(A(0, 0)\) and \(B(3, -1.5)\), the correct coordinates are \((2, -1)\), which corresponds to option c. Therefore, the coordinates of point \(P\) are \(\boxed{(2, -1)}\).
Examples
Architects often use the section formula when placing support pillars along a beam. If a beam spans from point A to point B, and they need to place a support at a position that is two-thirds of the distance from the wall (point A), they can use this formula to calculate the exact coordinate on their blueprint. If A is at \((0, 0)\) and B is at \((3, -1.5)\), placing the support at \(\frac{2}{3}\) of the way results in the coordinate \((2, -1)\), ensuring structural stability according to the design plan.