Explanation:

To determine whether the points (1, 2), (3, 8), and (x, 20) are collinear, we need to check if they lie on the same straight line.

Method 1: Slope of the Line

1. We can find the slope of the line passing through the first two points using the formula:

Slope = (y2 - y1) / (x2 - x1)

Substituting the coordinates of the given points, we have:

Slope = (8 - 2) / (3 - 1)
= 6 / 2
= 3

2. Now, we can check if the slope between the first and third points is also equal to 3.

Slope = (20 - 2) / (x - 1)
= 18 / (x - 1)

If the slope is equal to 3, then:

18 / (x - 1) = 3

Multiplying both sides by (x - 1), we get:

18 = 3(x - 1)
18 = 3x - 3
3x = 18 + 3
3x = 21
x = 21 / 3
x = 7

Since x = 7 satisfies the equation, the points (1, 2), (3, 8), and (7, 20) are collinear.

Method 2: Area of the Triangle

1. Another method to check collinearity is by calculating the area of the triangle formed by the three points.

2. We can use the formula for the area of a triangle using the coordinates of the points:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Substituting the coordinates of the given points, we have:

Area = 1/2 * |1(8 - 20) + 3(20 - 2) + x(2 - 8)|
= 1/2 * |-12 + 54 - 6x|
= 1/2 * |-12 + 54 - 6x|
= 1/2 * |42 - 6x|

3. For collinear points, the area of the triangle formed by them should be zero.

1/2 * |42 - 6x| = 0

|42 - 6x| = 0

42 - 6x = 0

6x = 42

x = 42 / 6

x = 7

Since x = 7 satisfies the equation, the points (1, 2), (3, 8), and (7, 20) are collinear.

Therefore, the correct answer is option 'D' (7).