Question

congruence stations use the distance formula to prove ab and cd are congruent.

Explanation:

Step1: Identify coordinates

From the graph, assume:

• \( A(1, -1) \), \( B(2, -3) \)
• \( C(-1, 2) \), \( D(-2, 0) \)

Step2: Apply distance formula

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

For \( AB \):

\( x_1 = 1, y_1 = -1 \); \( x_2 = 2, y_2 = -3 \)
\[

$$\begin{align*} AB &= \sqrt{(2 - 1)^2 + (-3 - (-1))^2} \\ &= \sqrt{(1)^2 + (-2)^2} \\ &= \sqrt{1 + 4} \\ &= \sqrt{5} \end{align*}$$

\]

For \( CD \):

\( x_1 = -1, y_1 = 2 \); \( x_2 = -2, y_2 = 0 \)
\[

$$\begin{align*} CD &= \sqrt{(-2 - (-1))^2 + (0 - 2)^2} \\ &= \sqrt{(-1)^2 + (-2)^2} \\ &= \sqrt{1 + 4} \\ &= \sqrt{5} \end{align*}$$

\]

Step3: Compare distances

Since \( AB = \sqrt{5} \) and \( CD = \sqrt{5} \), \( AB = CD \), so \( AB \) and \( CD \) are congruent.

Answer:

\( AB \) and \( CD \) are congruent (as their distances are both \(\sqrt{5}\)).