Given the vertices A(−3,4), B(8,4), and C(8,−7), which of the following is the final point needed to construct a square?
To construct a square with vertices A, B, and C, we need to find the fourth vertex.
Since the opposite sides of a square are parallel and of equal length, we can use the midpoint formula to find the midpoint of AB and BC. The midpoint formula states that the midpoint of two points (x₁, y₁) and (x₂, y₂) is given by:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The midpoint of AB is:
Midpoint AB = ((-3 + 8)/2, (4 + 4)/2) = (5/2, 8/2) = (2.5, 4)
The midpoint of BC is:
Midpoint BC = ((8 + 8)/2, (4 + (-7))/2) = (16/2, -3/2) = (8, -3/2)
Now, we can find the midpoint of the two found midpoints:
Midpoint = ((2.5 + 8)/2, (4 + (-3/2))/2) = (10.5/2, (8 - 3/2)/2) = (5.25, 5/2)
Therefore, the final point needed to construct a square is P(5.25, 2.5).