Answer :
To determine if segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are congruent, we need to calculate the lengths of both segments and compare them.
### Step 1: Calculate the Length of Segment [tex]\(\overline{AB}\)[/tex]
The coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are:
- [tex]\(A(3, -4)\)[/tex]
- [tex]\(B(-10, -4)\)[/tex]
The formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Using this formula for segment [tex]\(\overline{AB}\)[/tex]:
[tex]\[ \text{Length of } \overline{AB} = \sqrt{(-10 - 3)^2 + (-4 - (-4))^2} \][/tex]
Simplify inside the parentheses:
[tex]\[ = \sqrt{(-13)^2 + 0^2} \][/tex]
Calculate the squares:
[tex]\[ = \sqrt{169 + 0} \][/tex]
Finally, take the square root:
[tex]\[ = \sqrt{169} = 13 \][/tex]
So, the length of segment [tex]\(\overline{AB}\)[/tex] is 13.
### Step 2: Calculate the Length of Segment [tex]\(\overline{CD}\)[/tex]
The coordinates of points [tex]\(C\)[/tex] and [tex]\(D\)[/tex] are:
- [tex]\(C(-2, 5)\)[/tex]
- [tex]\(D(3, -7)\)[/tex]
Using the same distance formula for segment [tex]\(\overline{CD}\)[/tex]:
[tex]\[ \text{Length of } \overline{CD} = \sqrt{(3 - (-2))^2 + (-7 - 5)^2} \][/tex]
Simplify inside the parentheses:
[tex]\[ = \sqrt{(3 + 2)^2 + (-7 - 5)^2} \][/tex]
[tex]\[ = \sqrt{5^2 + (-12)^2} \][/tex]
Calculate the squares:
[tex]\[ = \sqrt{25 + 144} \][/tex]
Finally, take the square root:
[tex]\[ = \sqrt{169} = 13 \][/tex]
So, the length of segment [tex]\(\overline{CD}\)[/tex] is also 13.
### Step 3: Compare the Lengths
We found that:
- Length of [tex]\(\overline{AB}\)[/tex] = 13
- Length of [tex]\(\overline{CD}\)[/tex] = 13
Since both lengths are equal, the segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are congruent.
### Conclusion:
The segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are congruent since their lengths are equal.
### Step 1: Calculate the Length of Segment [tex]\(\overline{AB}\)[/tex]
The coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are:
- [tex]\(A(3, -4)\)[/tex]
- [tex]\(B(-10, -4)\)[/tex]
The formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Using this formula for segment [tex]\(\overline{AB}\)[/tex]:
[tex]\[ \text{Length of } \overline{AB} = \sqrt{(-10 - 3)^2 + (-4 - (-4))^2} \][/tex]
Simplify inside the parentheses:
[tex]\[ = \sqrt{(-13)^2 + 0^2} \][/tex]
Calculate the squares:
[tex]\[ = \sqrt{169 + 0} \][/tex]
Finally, take the square root:
[tex]\[ = \sqrt{169} = 13 \][/tex]
So, the length of segment [tex]\(\overline{AB}\)[/tex] is 13.
### Step 2: Calculate the Length of Segment [tex]\(\overline{CD}\)[/tex]
The coordinates of points [tex]\(C\)[/tex] and [tex]\(D\)[/tex] are:
- [tex]\(C(-2, 5)\)[/tex]
- [tex]\(D(3, -7)\)[/tex]
Using the same distance formula for segment [tex]\(\overline{CD}\)[/tex]:
[tex]\[ \text{Length of } \overline{CD} = \sqrt{(3 - (-2))^2 + (-7 - 5)^2} \][/tex]
Simplify inside the parentheses:
[tex]\[ = \sqrt{(3 + 2)^2 + (-7 - 5)^2} \][/tex]
[tex]\[ = \sqrt{5^2 + (-12)^2} \][/tex]
Calculate the squares:
[tex]\[ = \sqrt{25 + 144} \][/tex]
Finally, take the square root:
[tex]\[ = \sqrt{169} = 13 \][/tex]
So, the length of segment [tex]\(\overline{CD}\)[/tex] is also 13.
### Step 3: Compare the Lengths
We found that:
- Length of [tex]\(\overline{AB}\)[/tex] = 13
- Length of [tex]\(\overline{CD}\)[/tex] = 13
Since both lengths are equal, the segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are congruent.
### Conclusion:
The segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are congruent since their lengths are equal.
