Final answer:
To calculate the lengths and midpoints of the diagonals of the given rectangle, we use the distance formula to find that both diagonals have the same length, √181, and the midpoint formula reveals the midpoint of each to be (-1, -0.5).
Step-by-step explanation:
To find the lengths and midpoints of the diagonals of the rectangle with vertices (4,4), (4,-5), (-6,-5), (-6,4), we first identify that a rectangle has two congruent diagonals. The diagonal between the vertices (4,4) and (-6,-5) can be calculated using the distance formula
Length of diagonal = √[(x2 - x1)² + (y2 - y1)²]
= √[(-6 - 4)² + (-5 - 4)²]
= √[(-10)² + (-9)²]
= √[100 + 81]
= √181
The midpoint of this diagonal can be found using the midpoint formula, which is ((x1 + x2)/2, (y1 + y2)/2). Thus, the midpoint is ((4 - 6)/2, (4 - 5)/2) = (-1, -0.5). The other diagonal between the vertices (4,-5) and (-6,4) will have the same length and midpoint because they are congruent.