Question

find the distance between x and y with x(2, -9) and y(-10, -1)

Explanation:

Step1: Identify the coordinates

We have two points \( X(2, -9) \) and \( Y(-10, -1) \). Let \( (x_1, y_1) = (2, -9) \) and \( (x_2, y_2) = (-10, -1) \).

Step2: Apply the distance formula

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Substitute the values:
\( x_2 - x_1 = -10 - 2 = -12 \)
\( y_2 - y_1 = -1 - (-9) = -1 + 9 = 8 \)

Step3: Calculate the squares

\( (-12)^2 = 144 \)
\( 8^2 = 64 \)

Step4: Sum the squares

\( 144 + 64 = 208 \)

Step5: Take the square root

\( d = \sqrt{208} = \sqrt{16 \times 13} = 4\sqrt{13} \approx 14.42 \) (if we want a decimal approximation) or keep it as \( 4\sqrt{13} \). But let's check the calculation again for any missing part. Wait, maybe the y-coordinate of Y was missing? Wait, the original problem says "Y(-10, " maybe a typo, but assuming Y is (-10, -1) as per the keyboard (since there is -1). Wait, if we assume Y is (-10, -1), then the calculation is as above. Wait, but maybe the y-coordinate of Y is -1? Let's recheck.

Wait, the user's problem: "Find the distance between X and Y with X(2, -9) and Y(-10, " maybe a typo, but from the keyboard, there is a -1, so maybe Y is (-10, -1). So proceeding with that.

So, \( d = \sqrt{(-10 - 2)^2 + (-1 - (-9))^2} = \sqrt{(-12)^2 + (8)^2} = \sqrt{144 + 64} = \sqrt{208} = 4\sqrt{13} \approx 14.42 \). But if we simplify \( \sqrt{208} \), we can factor 208 as \( 16 \times 13 \), so \( \sqrt{16 \times 13} = 4\sqrt{13} \approx 14.42 \).

Wait, maybe the y-coordinate of Y is different? Wait, the original problem's Y is written as "Y(-10, " maybe a typo, but assuming it's Y(-10, -1) (since the keyboard has -1). So the distance is \( \sqrt{(-10 - 2)^2 + (-1 - (-9))^2} = \sqrt{(-12)^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} = 4\sqrt{13} \approx 14.42 \).

But let's confirm the formula again. The distance formula is correct. So steps:

1. Identify coordinates: \( X(2, -9) \), \( Y(-10, -1) \)
2. Apply distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
3. Substitute values: \( x_2 - x_1 = -12 \), \( y_2 - y_1 = 8 \)
4. Square the differences: \( (-12)^2 = 144 \), \( 8^2 = 64 \)
5. Sum: \( 144 + 64 = 208 \)
6. Take square root: \( \sqrt{208} = 4\sqrt{13} \approx 14.42 \)

Answer:

The distance between \( X \) and \( Y \) is \( 4\sqrt{13} \) (or approximately \( 14.42 \)). If we need an exact form, it's \( 4\sqrt{13} \), and if decimal, approximately \( 14.42 \).