Question

6.4 assignment
use distance to determine whether or not abcd is a parallelogram. (show both pairs of opposite sides congruent)
a (-2, -3) b (0, 5) c (6, 5) d (4, -3)
ab =
bc =
cd =
ad =
is \\(\overline{ab} \cong \overline{cd}\\)?
is \\(\overline{ad} \cong \overline{bc}\\)?
is abcd a parallelogram?

Explanation:

Step1: Recall distance formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Calculate \(AB\)

For \(A(-2,-3)\) and \(B(0,5)\):
\(AB=\sqrt{(0 - (-2))^2+(5 - (-3))^2}=\sqrt{(2)^2+(8)^2}=\sqrt{4 + 64}=\sqrt{68}=2\sqrt{17}\)

Step3: Calculate \(BC\)

For \(B(0,5)\) and \(C(6,5)\):
\(BC=\sqrt{(6 - 0)^2+(5 - 5)^2}=\sqrt{36+0}=\sqrt{36} = 6\)

Step4: Calculate \(CD\)

For \(C(6,5)\) and \(D(4,-3)\):
\(CD=\sqrt{(4 - 6)^2+(-3 - 5)^2}=\sqrt{(-2)^2+(-8)^2}=\sqrt{4 + 64}=\sqrt{68}=2\sqrt{17}\)

Step5: Calculate \(AD\)

For \(A(-2,-3)\) and \(D(4,-3)\):
\(AD=\sqrt{(4 - (-2))^2+(-3 - (-3))^2}=\sqrt{(6)^2+0}=\sqrt{36}=6\)

Step6: Compare \(AB\) and \(CD\)

Since \(AB = 2\sqrt{17}\) and \(CD=2\sqrt{17}\), \(\overline{AB}\cong\overline{CD}\) (yes).

Step7: Compare \(AD\) and \(BC\)

Since \(AD = 6\) and \(BC = 6\), \(\overline{AD}\cong\overline{BC}\) (yes).

Step8: Determine if parallelogram

A quadrilateral with both pairs of opposite sides congruent is a parallelogram, so \(ABCD\) is a parallelogram.

Answer:

\(AB = 2\sqrt{17}\)
\(BC = 6\)
\(CD = 2\sqrt{17}\)
\(AD = 6\)
Is \(\overline{AB}\cong\overline{CD}\)? Yes
Is \(\overline{AD}\cong\overline{BC}\)? Yes
Is \(ABCD\) a parallelogram? Yes