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 point \( (-1, 3) \) to the highway represented by the line passing through points \( (-4, -2) \) and \( (8, 4) \), we first need to determine the equation of the line connecting those points. 1. **Finding the slope**: The slope \( m \) of the line is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{8 - (-4)} = \frac{6}{12} = \frac{1}{2} \] 2. **Using point-slope form**: We can use one of the points, say \( (-4, -2) \), to write the equation of the line: \[ y - (-2) = \frac{1}{2}(x - (-4)) \] Simplifying this gives: \[ y + 2 = \frac{1}{2}(x + 4) \quad \Rightarrow \quad y = \frac{1}{2}x + 2 - 2 \quad \Rightarrow \quad y = \frac{1}{2}x - 2 \] 3. **Finding the perpendicular line from The Big Pumpkin**: The slope of the perpendicular line will be the negative reciprocal of \( \frac{1}{2} \), which is \( -2 \). Using point-slope form from the point \( (-1, 3) \): \[ y - 3 = -2(x + 1) \] Simplifying: \[ y - 3 = -2x - 2 \quad \Rightarrow \quad y = -2x + 1 \] 4. **Finding intersection point**: To find the shortest distance, we need the intersection of the two lines \( y = \frac{1}{2}x - 2 \) and \( y = -2x + 1 \). Setting them equal: \[ \frac{1}{2}x - 2 = -2x + 1 \] Multiplying through by 2 to clear the fraction: \[ x - 4 = -4x + 2 \quad \Rightarrow \quad 5x = 6 \quad \Rightarrow \quad x = \frac{6}{5} \] Substituting \( x = \frac{6}{5} \) back into one of the line equations to find \( y \): \[ y = \frac{1}{2} \cdot \frac{6}{5} - 2 = \frac{3}{5} - 2 = \frac{3}{5} - \frac{10}{5} = -\frac{7}{5} \] So the intersection point is \( \left( \frac{6}{5}, -\frac{7}{5} \right) \). 5. **Calculating the distance**: Now we can find the distance from \( (-1, 3) \) to \( \left( \frac{6}{5}, -\frac{7}{5} \right) \) using the distance formula: \[ d = \sqrt{ \left( -1 - \frac{6}{5} \right)^2 + \left( 3 - \left(-\frac{7}{5}\right) \right)^2 } \] Calculating each part: - \( -1 - \frac{6}{5} = -\frac{5}{5} - \frac{6}{5} = -\frac{11}{5} \) - \( 3 + \frac{7}{5} = \frac{15}{5} + \frac{7}{5} = \frac{22}{5} \) Plugging these into the distance formula: \[ d = \sqrt{ \left(-\frac{11}{5}\right)^2 + \left(\frac{22}{5}\right)^2 } \] \[ = \sqrt{ \frac{121}{25} + \frac{484}{25} } = \sqrt{ \frac{605}{25} } = \sqrt{24.2} \approx 4.9 \text{ km} \] To the nearest tenth of a kilometre, the shortest distance from The Big Pumpkin to the highway is approximately **4.9 km**.