Question

use coordinate geometry to prove congruence criteria. for any answers with a square root, write your answer in the form sqrt(___).

\\(ab = \\)
\\(bc = \sqrt{(2-3)^2 + (0-4)^2} = \\)
\\(ac = \sqrt{(0-3)^2 + (0-4)^2} = \\)
\\(de = \\)
\\(ef = \sqrt{((-1)-3)^2 + (2-1)^2} = \\)
\\(df = \sqrt{((-1)-3)^2 + (2-1-1))^2} = \\)

Explanation:

🆕 New Concept Discovered: Distance Formula in Coordinate Geometry
Measuring lengths on a grid using coordinates

Step 1: Identify the coordinates of the vertices

By analyzing the grid in the image:

• The origin \( (0,0) \) is marked by the intersection of the axes.
• Vertex \( A \) is at \( (0, 0) \).
• Vertex \( B \) is at \( (2, 3) \).
• Vertex \( C \) is at \( (3, 4) \).
• Vertex \( D \) is at \( (3, -1) \).
• Vertex \( E \) is at \( (3, 2) \).
• Vertex \( F \) is at \( (-1, 2) \).

Step 2: Calculate \( AB \)

Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for points \( A(0,0) \) and \( B(2,3) \):
\[ AB = \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{4 + 9} = \sqrt{13} \]

Following the requested format, this is written as:
sqrt(13)

Step 3: Calculate \( BC \)

Evaluate the given expression:
\[ BC = \sqrt{(2-3)^2 + (0-4)^2} \]
Note: Looking at the coordinates of \( B(2,3) \) and \( C(3,4) \), the standard distance formula is \( \sqrt{(3-2)^2 + (4-3)^2} = \sqrt{1+1} = \sqrt{2} \). However, evaluating the exact expression printed on the screen:
\[ \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \]

Following the requested format, this is written as:
sqrt(17)

Step 4: Calculate \( AC \)

Evaluate the given expression:
\[ AC = \sqrt{(0-3)^2 + (0-4)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Step 5: Calculate \( DE \)

Using the coordinates \( D(3,-1) \) and \( E(3,2) \):
Since they share the same x-coordinate, the distance is the vertical difference:
\[ DE = |2 - (-1)| = 3 \]

Step 6: Calculate \( EF \)

Evaluate the given expression:
\[ EF = \sqrt{((-1)-3)^2 + (2-1)^2} = \sqrt{(-4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

Following the requested format, this is written as:
sqrt(17)

Step 7: Calculate \( DF \)

Evaluate the given expression:
\[ DF = \sqrt{((-1)-3)^2 + (2-(-1))^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

Answer:

• \( AB \) = sqrt(13)
• \( BC \) = sqrt(17)
• \( AC \) = 5
• \( DE \) = 3
• \( EF \) = sqrt(17)
• \( DF \) = 5