Final answer:
To find the equation of the line through point A(5,7) that is perpendicular to the line segment AB, we first need to find the slope of AB. The slope of AB can be found using the formula (y2 - y1) / (x2 - x1). The equation of the line through A that is perpendicular to AB is y = (-1/2)x + 19/2, which can be written as 2y = -x + 19 when written in the form ax + by + c = 0.
Step-by-step explanation:
To find the equation of the line through point A(5,7) that is perpendicular to the line segment AB, we first need to find the slope of AB. The slope of AB can be found using the formula (y2 - y1) / (x2 - x1).
The coordinates of A are (5,7) and the coordinates of B are (-1,-5). Substituting these values into the formula, we get (7 - (-5)) / (5 - (-1)) = 12 / 6 = 2.
Since the line through A is perpendicular to AB, the slope of the line is the negative reciprocal of the slope of AB. Therefore, the slope of the line through A is -1/2.
Using the slope-intercept form of a straight line equation, y = mx + b, where m is the slope and b is the y-intercept, we can substitute the slope -1/2 and the coordinates of A (5,7) into the equation. y = (-1/2)x + b. To solve for b, we substitute the coordinates of A into the equation and solve for b. 7 = (-1/2)(5) + b. Simplifying the equation, we get 7 = -5/2 + b. Adding 5/2 to both sides, we get 7 + 5/2 = b. Common denominator of 2 for both sides become 14/2 + 5/2 = b. Simplifying, we get 19/2 = b.
Therefore, the equation of the line through A that is perpendicular to AB is y = (-1/2)x + 19/2, which can be written as 2y = -x + 19 when written in the form ax + by + c = 0.