Area Of Triangles And Quadrilaterals (Shoelace Formulae)

Explanation
In order to find the value of r, we need to calculate the area of the triangle using the given coordinates of the vertices. The formula for the area of a triangle is 1/2 * base * height. The base can be found by calculating the distance between points A and B, which is 4 units. The height can be found by calculating the distance between point C and the line AB, which is the perpendicular distance from point C to line AB. By using the formula for the distance between a point and a line, we can find that the height is 2 units. Therefore, the area of the triangle is 1/2 * 4 * 2 = 4 square units. Since the given area is 6 square units, it is not possible to find a value of r that satisfies this condition. Therefore, the answer is not available.

Explanation
To find the area of a triangle, we can use the formula: Area = 1/2 * base * height. In this case, we can take any two sides of the triangle as the base and height. Let's take AB as the base and the distance between AB and point C as the height. The distance between two points can be found using the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).

Using the distance formula, we find that the distance between AB and C is sqrt((4 - 2)^2 + (3 - (-3))^2) = sqrt(2^2 + 6^2) = sqrt(4 + 36) = sqrt(40) = 2sqrt(10).

Now, we can substitute the values into the area formula: Area = 1/2 * AB * height = 1/2 * sqrt((2 - (-2))^2 + (-3 - (-1))^2) * 2sqrt(10) = 1/2 * sqrt(4^2 + 2^2) * 2sqrt(10) = 1/2 * sqrt(16 + 4) * 2sqrt(10) = 1/2 * sqrt(20) * 2sqrt(10) = sqrt(20) * sqrt(10) = 2sqrt(2) * sqrt(10) = 2 * sqrt(2 * 10) = 2 * sqrt(20) = 2 * 2sqrt(5) = 4sqrt(5).

Therefore, the area of triangle ABC is 4sqrt(5) square units. However, the given answer is 14, which is incorrect.

Explanation
The correct answer is C. 1 : 3.

To find the area of triangle ABC, we can use the formula for the area of a triangle given three vertices. The coordinates of A, B, and C are given as (2, -3), (3, -1), and (2, 0) respectively. Using these coordinates, we can calculate the length of the base AB and the height from C to the line AB. Then, we can use the formula for the area of a triangle (1/2 * base * height) to find the area of triangle ABC.

Similarly, to find the area of triangle ACD, we can use the coordinates of A, C, and D which are (2, -3), (2, 0), and (-1, 1) respectively. By calculating the length of the base AC and the height from D to the line AC, we can find the area of triangle ACD using the same formula.

Comparing the two areas, we can see that the area of triangle ABC is 1/3 of the area of triangle ACD. Therefore, the ratio of the area of triangle ABC to the area of triangle ACD is 1 : 3.