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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?

Respuesta :

To prove that the diagonals are congruent, you need to formula to compute the distance between two points:

[tex] d(A,B) = \sqrt{(A_x-B_x)^2 + (A_y-B_y)^2} [/tex]

Using that formula, you may prove that [tex] d(C,E) = d(D,F) [/tex], which means that the two diagonals have the same length.

To prove that they are perpendicular, you need the formula to compute the slope of a segment. The slope, knowing the enpoints, is given by

[tex] m = \cfrac{\Delta y}{\Delta x} = \cfrac{A_y-B_y}{A_x-B_x} [/tex]

You can use this formula to prove that

[tex] m_{CE} = -\cfrac{1}{m_{DF}} [/tex]

In fact, if one slope is the opposite of the reciprocal of the other, the two segments are perpendicular.

Finally, to prove that they bisect each other, you first need to find the point where they meet. First of all, you need to find the line the segments lie on: the formula is

[tex] y-y_0 = m(x-x_0) [/tex]

where [tex] (x_0,y_0) [/tex] is one of the points belonging to the line, and you already know how to find the slope. Then, you find the point of intersection, say A, by solving the system involving the two lines:

[tex] \begin{cases} y= m_{CE}x+q_{CE}\\ y = m_{DF}x+q_{DF} \end{cases} [/tex]

And use again the formula for the distance between two points to prove that

[tex] d(A,C) = d(A,D) = d(A,E) = d(A,F) [/tex]

MrRoyal

The formulas that can prove that the diagonals are congruent perpendicular bisectors of each other are the distance formula and slope formula

The coordinates of the square are given as:

  • Point C = (1,1)
  • Point D = (3,1)
  • Point E = (3,-1)
  • Point F = (1,-1)

The diagonals of the square are:

CE and DF

So, we start by calculating the slopes of CD and DF using:

[tex]m = \frac{y_2 -y_1}{x_2 -x_1}[/tex]

So, we have:

[tex]m_{CE} = \frac{-1 -1}{3 -1}[/tex]

[tex]m_{CE} = \frac{-2}{2}[/tex]

[tex]m_{CE} = -1[/tex]

Also, we have:

[tex]m_{DF} = \frac{-1 -1}{1 -3}[/tex]

[tex]m_{DF} = \frac{-2}{-2}[/tex]

[tex]m_{DF} = 1[/tex]

Next, we calculate the distance of both diagonals using:

[tex]d = \sqrt{(x_2 -x_1)^2 + (y_2 - y_1)^2}[/tex]

So, we have:

[tex]CE = \sqrt{(3 -1)^2 + (-1 - 1)^2}[/tex]

[tex]CE = \sqrt{8}[/tex]

Also, we have:

[tex]DF = \sqrt{(3 -1)^2 + (1 -- 1)^2}[/tex]

[tex]DF = \sqrt{8}[/tex]

Notice that:

[tex]CE = DF = \sqrt{8}[/tex]

And the slopes of both lines are opposite reciprocals

Read more about distance and slope at:

https://brainly.com/question/13136492

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