Question

The vertices of a square CDEF are C(1,1), D(3,1), E(3,-1) and F(1,-1). What formulas prove that the diagonals are congruent perpendicular bisectors of each other?

Answer 1

The diagonals of the square CDEF are congruent perpendicular bisectors of each other. The length of diagonal CD is 2 and the length of diagonal EF is also 2. The diagonals intersect at the midpoint (2, 1).

Step-by-step explanation:

To prove that the diagonals of a square are congruent perpendicular bisectors of each other, we can use the distance formula and midpoint formula.

First, let's find the length of one of the diagonals. The length of diagonal CD can be found using the distance formula:

d = sqrt((x2-x1)^2 + (y2-y1)^2)

d = sqrt((3-1)^2 + (1-1)^2) = sqrt(4) = 2

Next, let's find the midpoint of diagonal CD using the midpoint formula:

midpoint = ((x1+x2)/2, (y1+y2)/2)

midpoint = ((1+3)/2, (1+1)/2) = (2, 1)

Now, let's find the length of the other diagonal EF using the same method:

d = sqrt((x2-x1)^2 + (y2-y1)^2)

d = sqrt((3-1)^2 + (-1-1)^2) = sqrt(4) = 2

Lastly, let's find the midpoint of diagonal EF:

midpoint = ((x1+x2)/2, (y1+y2)/2)

midpoint = ((1+3)/2, (-1-1)/2) = (2, -1)

Since the diagonals have the same length of 2 and intersect at the midpoint (2, 1), we can conclude that the diagonals are congruent perpendicular bisectors of each other.

Answer 2

To prove that the diagonals of a square are congruent perpendicular bisectors of each other, we can use the distance formula and midpoint formula.

Step-by-step explanation:

To prove that the diagonals of a square are congruent perpendicular bisectors of each other, we can use the distance formula and midpoint formula.

First, let's find the lengths of the diagonals. The diagonal CF has endpoints C(1,1) and F(1,-1), so its length can be found using the distance formula as CF = sqrt((1-1)^2 + (-1-1)^2) = sqrt(4) = 2. Similarly, the diagonal DE has endpoints D(3,1) and E(3,-1), so its length is DE = sqrt((3-3)^2 + (-1-1)^2) = sqrt(4) = 2.

Next, let's find the midpoints of the diagonals. The midpoint of CF is MCF = ((1+1)/2, (1-1)/2) = (1,0). The midpoint of DE is MDE = ((3+3)/2, (1-1)/2) = (3,0).

Since the lengths of the diagonals are equal (CF = DE) and the midpoints are equal (MCF = MDE), we can conclude that the diagonals are congruent and their midpoints are the same. Thus, the diagonals of the square CDEF are congruent perpendicular bisectors of each other.