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Given segments AB and CD intersect at E.
A(1, 2), B(4, 5), C(2, 4), D(2, 1), E(2, 3)

1. The length of AB in simplest radical form is ?

2. AB and CD are congruent or are not congruent?

3. AB bisects CD or AB does not bisect CD?

4. CD bisects AB or CD does not bisect AB?

Respuesta :

MrRoyal

The length of a segment is the distance between its endpoints.

  • [tex]\mathbf{AB = 3\sqrt{2}}[/tex]
  • AB and CD are not congruent
  • AB does not bisect CD
  • CD does not bisect AB

(a) Length of AB

We have:

[tex]\mathbf{A = (1,2)}[/tex]

[tex]\mathbf{B = (4,5)}[/tex]

The length of AB is calculated using the following distance formula

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

So, we have:

[tex]\mathbf{AB = \sqrt{(1 - 4)^2 + (2 - 5)^2}}[/tex]

[tex]\mathbf{AB = \sqrt{18}}[/tex]

Simplify

[tex]\mathbf{AB = 3\sqrt{2}}[/tex]

(b) Are AB and CD congruent

First, we calculate the length of CD using:

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

Where:

[tex]\mathbf{C = (2, 4)}[/tex]

[tex]\mathbf{D = (2, 1)}[/tex]

So, we have:

[tex]\mathbf{CD = \sqrt{(2 -2)^2 + (4 - 1)^2}}[/tex]

[tex]\mathbf{CD = \sqrt{9}}[/tex]

[tex]\mathbf{CD = 3}[/tex]

By comparison

[tex]\mathbf{CD \ne AB}[/tex]

Hence, AB and CD are not congruent

(c) AB bisects CD or not?

If AB bisects CD, then:

[tex]\mathbf{AB = \frac 12 \times CD}[/tex]

The above equation is not true, because:

[tex]\mathbf{3\sqrt 2 \ne \frac 12 \times 3}[/tex]

Hence, AB does not bisect CD

(d) CD bisects AB or not?

If CD bisects AB, then:

[tex]\mathbf{CD = \frac 12 \times AB}[/tex]

The above equation is not true, because:

[tex]\mathbf{3 \ne \frac 12 \times 3\sqrt 2}[/tex]

Hence, CD does not bisect AB

Read more about lengths and bisections at:

https://brainly.com/question/20837270