Given: The vertices of a triangle are (0, -1), (2, 1), and (0, 3).

To find: The area of the triangle formed by joining the mid-points of the sides of the given triangle and the ratio of this area to the area of the given triangle.

Solution:

Step 1: Find the mid-points of the sides of the given triangle.

The mid-point of the line segment joining the points (x1, y1) and (x2, y2) is given by:

((x1 + x2)/2, (y1 + y2)/2)

Using this formula, we can find the mid-points of the sides of the given triangle:

Mid-point of AB: ((0 + 2)/2, (-1 + 1)/2) = (1, 0)

Mid-point of BC: ((2 + 0)/2, (1 + 3)/2) = (1, 2)

Mid-point of AC: ((0 + 0)/2, (-1 + 3)/2) = (0, 1)

Step 2: Join the mid-points to form a triangle.

We join the mid-points (1, 0), (1, 2), and (0, 1) to form a triangle.

Step 3: Find the area of the triangle formed by the mid-points.

We can find the area of the triangle formed by the mid-points using the formula:

Area = 1/2 * base * height

The base of the triangle is the distance between the mid-points (1, 0) and (1, 2), which is 2 units.

The height of the triangle is the distance between the mid-point (0, 1) and the line containing the base. We can find this distance using the formula for the distance between a point and a line:

Distance = |ax + by + c| / sqrt(a^2 + b^2)

where the line is ax + by + c = 0.

The equation of the line containing the base is x = 1. Substituting this into the formula, we get:

Distance = |1(0) + 1(1) - 1| / sqrt(1^2 + 1^2) = 1 / sqrt(2)

Therefore, the height of the triangle is 1 / sqrt(2) units.

Hence, the area of the triangle formed by the mid-points is:

Area = 1/2 * base * height = 1/2 * 2 * 1/sqrt(2) = sqrt(2)

Step 4: Find the area of the given triangle.

We can find the area of the given triangle using the formula:

Area = 1/2 * base * height

where the base is the distance between any two vertices, and the height is the perpendicular distance from the third vertex to the line containing the base.

Let's take AB as the base. The distance between A(0, -1) and B(2, 1) is:

sqrt((2 - 0)^2 + (1 - (-1))^2) = sqrt(20)

The perpendicular distance from C(0, 3) to