Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary. (1 point) Responses 5.57 5.57 41 41 6:40 6:40 9

To use the Pythagorean Theorem to find the length between two points, you would typically have two points in a coordinate system, such as (x₁, y₁) and (x₂, y₂). The formula to find the distance \( d \) between these two points is given by:

\[ d = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²} \]

If you can provide the coordinates of the two points, I can help calculate the distance for you.

To find the distance between the points \((4, 2)\) and \((-2, -1)\), we can apply the Pythagorean Theorem using the distance formula:

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

Here, \((x_1, y_1) = (4, 2)\) and \((x_2, y_2) = (-2, -1)\).

Now, we can plug in the coordinates:

\[ d = \sqrt{((-2) - (4))^2 + ((-1) - (2))^2} \]

Calculating the differences:

\[ d = \sqrt{(-6)^2 + (-3)^2} \]

Now square the differences:

\[ d = \sqrt{36 + 9} \]

Adding the squared values:

\[ d = \sqrt{45} \]

Finally, we can simplify \(\sqrt{45}\):

\[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]

Calculating this value:

\[ 3\sqrt{5} \approx 3 \times 2.236 \approx 6.708 \]

Rounding to the nearest hundredth gives:

\[ \approx 6.71 \]

So, the distance between the points \((4, 2)\) and \((-2, -1)\) is approximately 6.71.

No problem! Let's find the distance between the points \((2, 2)\) and \((-1, -2)\) using the distance formula:

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

Here, \((x_1, y_1) = (2, 2)\) and \((x_2, y_2) = (-1, -2)\).

Now, we can plug in the coordinates:

\[ d = \sqrt{((-1) - (2))^2 + ((-2) - (2))^2} \]

Calculating the differences:

\[ d = \sqrt{(-3)^2 + (-4)^2} \]

Now square the differences:

\[ d = \sqrt{9 + 16} \]

Adding the squared values:

\[ d = \sqrt{25} \]

Taking the square root gives:

\[ d = 5 \]

So, the distance between the points \((2, 2)\) and \((-1, -2)\) is 5.

Got it! Let's find the distance between the points \((2, 4)\) and \((-1, -2)\) using the distance formula:

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

Here, \((x_1, y_1) = (2, 4)\) and \((x_2, y_2) = (-1, -2)\).

Now we plug in the coordinates:

\[ d = \sqrt{((-1) - (2))^2 + ((-2) - (4))^2} \]

Calculating the differences:

\[ d = \sqrt{(-3)^2 + (-6)^2} \]

Now square the differences:

\[ d = \sqrt{9 + 36} \]

Adding the squared values:

\[ d = \sqrt{45} \]

We can simplify \( \sqrt{45} \):

\[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \]

Calculating this value:

\[ 3\sqrt{5} \approx 3 \times 2.236 \approx 6.708 \]

Rounding to the nearest hundredth gives:

\[ \approx 6.71 \]

So, the distance between the points \((2, 4)\) and \((-1, -2)\) is approximately 6.71.

Based on the calculation, the distance between the points \((2, 4)\) and \((-1, -2)\) is approximately 6.71.

Since 6.71 does not appear in the options you've provided, the closest available options are 5.57, 41, 6:40, or 9. If you were looking for a numerical response from that set of options, none would accurately reflect the calculated distance.

Please double-check the options or confirm if there might have been a miscommunication regarding the question or data.