Show that the following points are collinear:
A (5,2)
B (-3,-2)
C (9,4)
they all satisfy
x-2y = 1
To determine whether the given points A (5,2), B (-3,-2), and C (9,4) are collinear, we can use the concept of slopes. If the slopes of any two pairs of points are equal, then the points are collinear.
Step 1: Find the slope of line AB
Let's find the slope between points A and B using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates of A (5,2) and B (-3,-2) into the formula, we get:
m_AB = (-2 - 2) / (-3 - 5) = -4 / -8 = 1/2
Step 2: Find the slope of line AC
Now, let's find the slope between points A and C using the same formula:
m_AC = (4 - 2) / (9 - 5) = 2 / 4 = 1/2
Step 3: Compare the slopes
The slopes of lines AB and AC are both equal to 1/2. Therefore, we can conclude that the points A, B, and C are collinear.
Alternatively, if you have access to graphing software or a graphing calculator, you can plot the points A, B, and C and visually determine whether they lie on the same line. If the points appear to be in a straight line, they are collinear.