Short Answer 
The equation of a straight line is given by ( y = mx + c ), where ( m ) is the slope and ( c ) is the y-intercept. For line BC through points B(7,5) and C(2,3), the slope is calculated as ( 2/5 ). The perpendicular line passing through point A has a slope of (-5/2) and the final equation is ( 2y = -5x + 31 ).
Step 1: Understand the Equation of a Straight Line
The general form of the equation for a straight line is represented as y = mx + c. Here, m stands for the slope of the line, which indicates how steep the line is, while c is the y-intercept, the point where the line crosses the y-axis. Understanding these components is crucial for defining any line on a Cartesian plane.
Step 2: Calculate the Slope of Line BC
To find the equation of line BC that passes through points B(7,5) and C(2,3), we first determine its slope. The slope (m) can be calculated using the formula:
- m = (y2 – y1) / (x2 – x1)
Substituting the coordinates, we find:
- m = (3 – 5) / (2 – 7) = 2/5
Next, we can find the entire equation of line BC using the point-slope form of the equation.
Step 3: Determine the Perpendicular Line Through Point A
To find the equation of a line that is perpendicular to line BC and passes through a given point A, we need to use the negative reciprocal of the slope of line BC. Since the slope of BC is 2/5, the slope of the perpendicular line is -5/2. Using point A’s coordinates and this slope, we can now formulate the equation:
- Substituting into the equation: y = (-5/2)x + c
- Calculate c using point A’s coordinates to finalize the equation.
Thus, the resulting equation for the line perpendicular to BC passing through A is 2y = -5x + 31.