Question

in the coordinate plane, plot $overline{ab}$ and $overline{cd}$ given by the points $a(8,3)$, $b(-1,3)$, $c(5,10)$, $d(5,3)$. determine whether $overline{ab}$ and $overline{cd}$ are congruent.

Explanation:

Step1: Recall distance - formula

The distance formula 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 the length of $\overline{AB}$

For points $A(8,3)$ and $B(-1,3)$, since $y_1 = y_2=3$, then $d_{AB}=\sqrt{( - 1 - 8)^2+(3 - 3)^2}=\sqrt{(-9)^2+0^2}=\sqrt{81}=9$.

Step3: Calculate the length of $\overline{CD}$

For points $C(5,10)$ and $D(5,3)$, since $x_1 = x_2 = 5$, then $d_{CD}=\sqrt{(5 - 5)^2+(10 - 3)^2}=\sqrt{0^2+7^2}=\sqrt{49}=7$.

Step4: Compare the lengths

Since $d_{AB}=9$ and $d_{CD}=7$, and $9
eq7$.

Answer:

$\overline{AB}$ and $\overline{CD}$ are not congruent.