Final answer:
To determine the coordinates of the third point for the given points to be collinear, calculate the slope between the first two points and then substitute the answer choices to check which coordinates satisfy the equation. The correct answer is (9, 5).
Step-by-step explanation:
To determine the coordinates of the third point for the given points to be collinear, we need to determine if the slope between the first two points is the same as the slope between the second and third points.
First, let's calculate the slope between the first two points:
Slope = (y2 - y1) / (x2 - x1) = (3 - 1) / (6 - 3) = 2 / 3
Now, let's plug in the coordinates of the third point and calculate its slope with the second point:
Slope = (y3 - y2) / (x3 - x2) = (y3 - 3) / (x3 - 6)
Since the two slopes should be equal, we can set up the equation (2/3) = (y3 - 3) / (x3 - 6), and solve for y3 in terms of x3:
2/3 = (y3 - 3) / (x3 - 6)
2(x3 - 6) = 3(y3 - 3)
2x3 - 12 = 3y3 - 9
3y3 = 2x3 - 3
y3 = (2/3)x3 - 1
So, the coordinates of the third point for the points to be collinear are (x3, y3) = (x3, (2/3)x3 - 1).
Now, let's substitute the answer choices and check which coordinates satisfy the equation.
a) (9, 5): y3 = (2/3) * 9 - 1 = 6 - 1 = 5, so this satisfies the equation.
b) (1, 3): y3 = (2/3) * 1 - 1 = 2/3 - 1 = -1/3, so this doesn't satisfy the equation.
c) (0, 0): y3 = (2/3) * 0 - 1 = -1, so this doesn't satisfy the equation.
d) (4, 2): y3 = (2/3) * 4 - 1 = 8/3 - 1 = 5/3, so this doesn't satisfy the equation.
Therefore, the correct answer is a) (9, 5).