Question
Find the distance between the points. Give an exact answer and an approximation to three decimal places. \( (-1,-6) \) and \( (-11,-11) \)
Ask by Lambert Ruiz. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The distance between the points \((-1, -6)\) and \((-11, -11)\) is exactly \(5\sqrt{5}\) units, which is approximately 11.180 units.
Solution
To find the distance between the points \((-1, -6)\) and \((-11, -11)\), we can use the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the given coordinates:
\[
x_1 = -1, \quad y_1 = -6, \quad x_2 = -11, \quad y_2 = -11
\]
Calculate the differences:
\[
x_2 - x_1 = -11 - (-1) = -10
\]
\[
y_2 - y_1 = -11 - (-6) = -5
\]
Now, substitute these into the distance formula:
\[
\text{Distance} = \sqrt{(-10)^2 + (-5)^2} = \sqrt{100 + 25} = \sqrt{125}
\]
**Exact Answer:**
\[
5\sqrt{5} \text{ units}
\]
**Approximation to Three Decimal Places:**
\[
5\sqrt{5} \approx 5 \times 2.236 = 11.180 \text{ units}
\]
**Final Answer:**
The distance is exactly \(5\sqrt{5}\) units and approximately 11.180 units.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the distance between the points \((-1,-6)\) and \((-11,-11)\), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in our points: \[ d = \sqrt{((-11) - (-1))^2 + ((-11) - (-6))^2} \] \[ d = \sqrt{(-11 + 1)^2 + (-11 + 6)^2} \] \[ d = \sqrt{(-10)^2 + (-5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \] The exact answer is \(5\sqrt{5}\). To approximate this, we can calculate \(5 \times 2.236\) (since \(\sqrt{5} \approx 2.236\)): \[ 5\sqrt{5} \approx 5 \times 2.236 \approx 11.180 \] Thus, the distance between the points is \(5\sqrt{5}\) (exact) and approximately \(11.180\) (to three decimal places).
