Two vertices of a triangle are F (1, −4) and G(6, −4). List two possible coordinates of the third vertex so that the triangle has an area of 20 square units.
To find two possible coordinates of the third vertex of the triangle, we need to first determine the base and the height of the triangle. The base can be the horizontal distance between points F and G, and the height can be the vertical distance between the base and the third vertex.
Step 1: Find the length of the base.
The base is the horizontal distance between points F(1, -4) and G(6, -4). To find the length of the base, we subtract the x-coordinate of F from the x-coordinate of G.
Base = Gx - Fx = 6 - 1 = 5
Step 2: Calculate the height.
The area of the triangle is given as 20 square units. The area of a triangle can be calculated using the formula: Area = (1/2) * base * height. Since we know the area and the base, we can rearrange the formula to solve for the height:
Height = (2 * Area) / base = (2 * 20) / 5 = 40 / 5 = 8
Step 3: Determine the y-coordinate of the third vertex.
Since the base is horizontal, the height will be vertical. We want the third vertex to be on the same horizontal line as F and G to form a triangle with an area of 20 square units. Given that the y-coordinate of F and G is -4, the y-coordinate of the third vertex will also be -4 ± height = -4 ± 8.
Therefore, two possible coordinates for the third vertex are...
1. Third vertex = H(1, -4 + 8) = H(1, 4)
2. Third vertex = I(1, -4 - 8) = I(1, -12)
So, the two possible coordinates of the third vertex are H(1,4) and I(1,-12).