Using the three indicated points on the given line and the concept of similar triangles, show and explain why the slope of the line is the same between any two of these three points. Calculate the slope of the line.
The three indicated points on the given line are P(1,2), Q(3,4), and R(5,6).
To show that the slope of the line is the same between any two of these three points using similar triangles, we can consider the triangles formed by these three points and the y-intercept of the line.
Let's consider the triangle formed by points P(1,2), Q(3,4), and the y-intercept (0,b) of the line. The slope of the line passing through points P and Q can be calculated as:
m1 = (4-2)/(3-1) = 2/2 = 1
Now, let's consider the triangle formed by points Q(3,4), R(5,6), and the y-intercept (0,b) of the line. The slope of the line passing through points Q and R can be calculated as:
m2 = (6-4)/(5-3) = 2/2 = 1
Since the slopes m1 and m2 are the same, we can conclude that the slope of the line is the same between any two of these three points.
To calculate the slope of the line passing through points P(1,2) and Q(3,4), we can use the formula:
m = (y2-y1)/(x2-x1) = (4-2)/(3-1) = 2/2 = 1
Therefore, the slope of the line passing through points P(1,2) and Q(3,4) is 1.