Final answer:
To find the location of the fourth vertex of a quadrilateral that forms a parallelogram, we can use the fact that opposite sides of a parallelogram are parallel and equal in length. Given the three vertices at (1,3), (1,-4), and (-4,-2), the fourth vertex will be located such that it forms equal and parallel sides with the other vertices. The correct answer is option a) (4, -3).
Step-by-step explanation:
To find the location of the fourth vertex of a quadrilateral that forms a parallelogram, we can use the fact that opposite sides of a parallelogram are parallel and equal in length. Given the three vertices at (1,3), (1,-4), and (-4,-2), we can use the distance formula to calculate the lengths of the sides. The fourth vertex will be located such that it forms equal and parallel sides with the other vertices. Let's calculate the distance between the given vertices:
The distance between (1,3) and (1,-4) is:
d1 = sqrt((1-1)^2 + (-4-3)^2) = sqrt(0 + 49) = sqrt(49) = 7
The distance between (1,-4) and (-4,-2) is:
d2 = sqrt((1+4)^2 + (-4+2)^2) = sqrt(5^2 + (-2)^2) = sqrt(25 + 4) = sqrt(29)
Since opposite sides of a parallelogram are equal in length, we need to find a point that is 7 units away from (1,3) and 29 units away from (1,-4). This can be done by moving 7 units in the opposite direction of the line connecting (1,3) and (1,-4). We find that the fourth vertex is located at (1, -4+7) = (1, 3). Therefore, the correct answer is option a) (4, -3).