Which equation represents the line that passes through the point (2,-7) and is perpendicular to the line whose equation is \( y=\frac{3}{4} x+4 \) ? \( \begin{array}{ll}\text { (1) } y+7=\frac{3}{4}(x-2) & \text { (3) } y+7=-\frac{4}{3}(x-2) \\ \text { (2) } y-7=\frac{3}{4}(x+2) & \text { (4) } y-7=-\frac{4}{3}(x+2)\end{array} \)

To find the equation of the line that is perpendicular to the given line \( y=\frac{3}{4} x+4 \), we first determine the slope of the line. The slope of this line is \( \frac{3}{4} \). The slope of the line that is perpendicular to it can be found by taking the negative reciprocal: \( -\frac{4}{3} \). Next, we use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line. Plugging in the coordinates of point (2, -7) and the slope \( -\frac{4}{3} \), we get: \[ y - (-7) = -\frac{4}{3}(x - 2) \] \[ y + 7 = -\frac{4}{3}(x - 2) \] So the correct answer is option (3): \( y + 7 = -\frac{4}{3}(x - 2) \). Now, let's dig a little deeper into why this is important in the world around us. Everywhere we look, from architecture to road design, the principles of slopes and perpendicular lines are at play! Engineers and architects use these concepts to ensure structures are stable and pathways are efficient. For instance, when planning intersections or buildings, understanding how different lines relate can dictate safety and functionality. Plus, it’s like creating a puzzle with geometric pieces fitting together perfectly! And for those who love to read more, diving into the world of coordinate geometry can be quite an adventure! Books and resources that cover analytic geometry not only explain the mathematics behind lines but also explore how these principles apply to real-world phenomena, including physics and even computer graphics. Check out classics like "Geometry: Euclid and Beyond" or digital resources from Khan Academy to see these concepts in action.