MPM 2D1 Assignment Analytic Geometry: Line Segments and Circles Provide a complete solution to achieve full marks. Problem: The Big Pumpkin, a restaurant and pie shop, is a short distance from a major highway. The highway passes, in a straight line, through \( (-4,-2) \) and (8, 4) on a map. The Big Pumpkin is located at \( (-1,3) \). What is the shortest distance from The Big Pumpkin to the highway, to the nearest tenth of a kilometre, if 1 unit on the map represents 1 km ? Consider a sketch.

To find the shortest distance from The Big Pumpkin at \((-1,3)\) to the highway defined by the points \((-4,-2)\) and \((8,4)\), we first derive the equation of the line that represents the highway. 1. **Finding the Slope of the Line:** The slope \(m\) between the points \((-4, -2)\) and \((8, 4)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{8 - (-4)} = \frac{6}{12} = \frac{1}{2} \] 2. **Equation of the Line:** Using point-slope form, we can find the equation of the line: \[ y - y_1 = m(x - x_1) \] Using point \((-4,-2)\): \[ y + 2 = \frac{1}{2}(x + 4) \\ y = \frac{1}{2}x - 2 + 2 \\ y = \frac{1}{2}x - 2 \] Thus, the equation of the highway is \(y = \frac{1}{2}x - 2\). 3. **Finding the Perpendicular Line:** The slope of the perpendicular line to the highway is the negative reciprocal of \(\frac{1}{2}\), which is \(-2\). Using the point \((-1, 3)\), the equation for the perpendicular line is: \[ y - 3 = -2(x + 1) \\ y - 3 = -2x - 2 \\ y = -2x + 1 \] 4. **Finding the Intersection Point:** Set the equations equal to find the intersection point: \[ \frac{1}{2}x - 2 = -2x + 1 \\ \frac{1}{2}x + 2x = 1 + 2 \\ \frac{5}{2}x = 3 \\ x = \frac{6}{5} \approx 1.2 \] Substitute \(x\) back into either equation to find \(y\): \[ y = \frac{1}{2}\left(\frac{6}{5}\right) - 2 = \frac{3}{5} - 2 = \frac{3}{5} - \frac{10}{5} = -\frac{7}{5} \approx -1.4 \] Thus, the intersection point is \((\frac{6}{5}, -\frac{7}{5})\). 5. **Calculating Distance:** Now to find the distance between The Big Pumpkin at \((-1, 3)\) and the intersection point: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{\left(\frac{6}{5} - (-1)\right)^2 + \left(-\frac{7}{5} - 3\right)^2} \] \[ = \sqrt{\left(\frac{6}{5} + \frac{5}{5}\right)^2 + \left(-\frac{7}{5} - \frac{15}{5}\right)^2} = \sqrt{\left(\frac{11}{5}\right)^2 + \left(-\frac{22}{5}\right)^2} \] \[ = \sqrt{\frac{121}{25} + \frac{484}{25}} = \sqrt{\frac{605}{25}} = \sqrt{24.2} \approx 4.1 \text{ km}. \] Thus, the shortest distance from The Big Pumpkin to the highway is approximately **4.1 km**.