Question

find the distance between the pair of points (5.3,8.2) and (-0.7,5.2). if necessary, express the answer in simplified radical form and then round to two decimal places.
the distance between the given points is □ units.
(simplify your answer. type an exact answer, using radicals as needed.)
express the distance rounded to two decimal places if the exact simplified answer contains radicals.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the distance between the given points is approximately □ units.
(type an integer or decimal rounded to two decimal places as needed.)
b. the exact simplified form does not contain radicals.

Explanation:

Step1: Apply the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 5.3\), \(y_1=8.2\), \(x_2=-0.7\), \(y_2 = 5.2\).
Substitute the values: \(d=\sqrt{(-0.7 - 5.3)^2+(5.2 - 8.2)^2}\)

Step2: Simplify the expressions inside the square - root

First, simplify \(-0.7-5.3=-6\) and \(5.2 - 8.2=-3\).
Then \(d=\sqrt{(-6)^2+(-3)^2}=\sqrt{36 + 9}\)

Step3: Calculate the sum inside the square - root

\(36+9 = 45\), so \(d=\sqrt{45}\)

Step4: Simplify the radical

\(\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}\)

Step5: Approximate the value

\(3\sqrt{5}\approx3\times2.236 = 6.708\approx6.71\)

Answer:

The distance between the given points is \(3\sqrt{5}\) units.
A. The distance between the given points is approximately \(6.71\) units.