Find the distance between the pair of points $$ (5,5) $$ and $$ (9,9) $$. If necessary, express the answer in simplified radical form and then round to two decimal places. The distance between the given points is $$ \square $$ 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 $$ \square $$ units. (Type an integer or decimal rounded to two decimal places as needed.) B. The exact simplified form does not contain radicals.

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UpStudy AI · Accepted Answer

To find the distance between the points $$ (5,5) $$ and $$ (9,9) $$, we can use the distance formula:

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

Here, $$ (x_1, y_1) = (5, 5) $$ and $$ (x_2, y_2) = (9, 9) $$.

Substituting the coordinates into the formula:

$$
d = \sqrt{(9 - 5)^2 + (9 - 5)^2}
$$

Calculating the differences:

$$
d = \sqrt{(4)^2 + (4)^2}
$$

Calculating the squares:

$$
d = \sqrt{16 + 16}
$$

Adding the values:

$$
d = \sqrt{32}
$$

We can simplify $$ \sqrt{32} $$:

$$
\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}
$$

Now, we can also calculate the approximate value of $$ 4\sqrt{2} $$:

Using $$ \sqrt{2} \approx 1.414 $$:

$$
4\sqrt{2} \approx 4 \times 1.414 \approx 5.656
$$

Rounding to two decimal places, we get:

$$
5.66
$$

Thus, the distance between the given points is:

- Exact simplified form: $$ 4\sqrt{2} $$
- Approximate value rounded to two decimal places: $$ 5.66 $$

The final answer is:

A. The distance between the given points is approximately $$ 5.66 $$ units.
The distance is $$ 4\sqrt{2} $$ units or approximately $$ 5.66 $$ units.

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