Answers plus Step-by-step explanations:
The graph of the line passes through the points (0,6) and (3,0)
(a) The equation of the line AB:
The slope of the line is;
Slope = change in y ÷ change in x = [tex]\frac{0 - 6}{3 - 0}[/tex] = -2
Picking another point (x,y) on the line;
Slope = [tex]\frac{y - 0}{x - 3}[/tex] = -2
Cross-multiplying gives;
y = -2x + 6 (which is the equation of line AB)
(b) The gradient of the line perpendicular to AB:
The products of slopes of two perpendicular lines = -1
Since the slope of our line = -2,
We assume the perpendicular line has a slope of a,
So a × -2 = -1
a = [tex]\frac{1}{2}[/tex]
So the slope (gradient) of the perpendicular line = [tex]\frac{1}{2}[/tex]
(c) The equation of the line passing through point A and perpendicular to AB:
Point A is the point (0,6) on the graph.
A line that passes through this point and is perpendicular to AB must have a gradient (or slope) of [tex]\frac{1}{2}[/tex]
Taking another point (x,y) on the perpendicular line;
Slope = change in y ÷ change in x
[tex]\frac{1}{2}[/tex] = [tex]\frac{y - 6}{x - 0}[/tex]
Cross-multiplying gives;
2y - 12 = x
2y = x + 12
y = [tex]\frac{x}{2}[/tex] + 6 (the equation of the perpendicular line via A)
