Question
What is the distance between the points \( (5,7) \) and \( (-1,11) \) ? Give your answer in radical form. \( d=\square \) units
Ask by Vargas Harris. in the United States
Jan 15,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The distance between the points \( (5,7) \) and \( (-1,11) \) is \( 2\sqrt{13} \) units.
Solution
To find the distance between the points \( (5, 7) \) and \( (-1, 11) \), we can use the **distance formula**:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
**Step-by-Step Calculation:**
1. **Identify the coordinates:**
- Point 1: \( (x_1, y_1) = (5, 7) \)
- Point 2: \( (x_2, y_2) = (-1, 11) \)
2. **Calculate the differences:**
\[
x_2 - x_1 = -1 - 5 = -6
\]
\[
y_2 - y_1 = 11 - 7 = 4
\]
3. **Square the differences:**
\[
(-6)^2 = 36
\]
\[
4^2 = 16
\]
4. **Add the squared differences:**
\[
36 + 16 = 52
\]
5. **Take the square root to find the distance:**
\[
d = \sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}
\]
**Final Answer:**
\[
d = \boxed{2\sqrt{13}} \text{ units}
\]
Reviewed and approved by the UpStudy tutoring team
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To find the distance between the points \( (5,7) \) and \( (-1,11) \), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates \( (x_1, y_1) = (5, 7) \) and \( (x_2, y_2) = (-1, 11) \): \[ d = \sqrt{((-1) - 5)^2 + (11 - 7)^2} \] \[ d = \sqrt{(-6)^2 + (4)^2} \] \[ d = \sqrt{36 + 16} \] \[ d = \sqrt{52} \] Simplifying \(\sqrt{52}\): \[ d = \sqrt{4 \times 13} = 2\sqrt{13} \] Thus, the distance \( d = 2\sqrt{13} \) units.